3. Matrix Algebra
• Matrix algebra is a means of expressing large
numbers of calculations made upon ordered sets of
numbers.
• Often referred to as Linear Algebra
• Many equations would be completely intractable if
scalar mathematics had to be used. It is also
important to note that the scalar algebra is under
there somewhere.
4. Matrix (Basic Definitions)
An m × n matrix A is a rectangular array of
numbers with m rows and n columns. (Rows are
horizontal and columns are vertical.) The numbers
m and n are the dimensions of A. The numbers in
the matrix are called its entries. The entry in row i
and column j is called aij .
a11 , , a1n
a21 , , a2 n
A Aij
ak 1 , , akn
4
5. Matrix
A matrix is any doubly subscripted array of
elements arranged in rows and columns.
a11 , , a1n
a 21 , , a 2n
A Aij
am1 , , amn
6. Definitions - Matrix
• A matrix is a set of rows and columns of
numbers 1 2 3
4 5 6
• Denoted with a bold Capital letter
• All matrices (and vectors) have an order -
that is the number of rows x the number of
columns.
• Thus A = 1 2 3
4 5 6 2x3
7. Definitions - scalar
• scalar - a number
– denoted with regular type as is scalar algebra
– [1] or [a]
8. Definitions - vector
• vector - a single row or column of numbers
– denoted with bold small letters
– row vector a = 1 2 3 4 5
– column vector x =
x1
x2
x3
x4
x5
11. Special matrices
• There are a number of special matrices
– Square
– Diagonal
– Symmetric
– Null
– Identity
12. Square matrix
• A square matrix is just what it sounds like, an nxn matrix
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
• Square matrices are quite useful for describing the
properties or interrelationships among a set of things –
like a data set.
14. Diagonal Matrices
– A diagonal matrix is a square matrix that has
values on the diagonal with all off-diagonal
entities being zero.
a11 0 0 0
0 a22 0 0
0 0 a33 0
0 0 0 a44
15. Symmetric Matrix
• All of the elements in the upper right portion of
the matrix are identical to those in the lower
left.
• For example, the correlation matrix
16. Identity Matrix
• The identity matrix I is a diagonal matrix
where the diagonal elements all equal one.
It is used in a fashion analogous to multiplying
through by "1" in scalar math.
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
17. Null Matrix
• A square matrix where all elements equal zero.
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
• Not usually ‘used’ so much as sometimes the result
of a calculation.
– Analogous to “a+b=0”
18. Types of Matrix
• Identity matrices - I • Symmetric
1 0 0 0 ab c
10 0 1 0 0 bd e
01 0 0 1 0 c e f
0 0 0 1 – Diagonal matrices are (of
• Diagonal course) symmetric
– Identity matrices are (of
1 0 0 0 course) diagonal
0 2 0 0
0 0 1 0
0 0 0 4
21. Operations with Matrices (Transpose)
Transpose
The transpose, AT , of a matrix A is the matrix obtained from A by
writing its rows as columns. If A is an k×n matrix and B = AT then
B is the n×k matrix with bij = aji. If AT=A, then A is symmetric.
Example:
T a11 a21
a11 a12 a13
a12 a22
a21 a22 a23
a13 a23
It it easy to verify :
(A B)T AT B T , (A B)T AT BT ,
(AT )T A, (rA)T rAT
w here A and B are k n and r is a scalar.
Let C be a k m matrix and D be an m n matrix. Then,
(CD)T DT C T ,
22. The Transpose of a Matrix At
• Taking the transpose is an operation that
creates a new matrix based on an existing
one.
• The rows of A = the columns of At
• Hold upper left and lower right corners and
rotate 180 degrees.
23. Transpose Matrix
Rows become columns and
columns become rows
a11 , a12, , a1n a11 , a 21 , , an1
a 21 , a 22, , a2n t a12 , a 22 , , an 2
A A
am1 , am 2 , amn a1m, a 2 m , , anm
24. Example of a transpose
1 4
t 1 2 3
A 2 5 ,A
4 5 6
3 6
25. The Transpose of a Matrix At
• If A = At, then A is symmetric (i.e. correlation matrix)
• If A AT = A then At is idempotent
– (and A' = A)
• The transpose of a sum = sum of transposes
• The transpose of a product = the product of the
transposes in reverse order
(A B C)t At Bt Ct
27. Transpose Matrix
Ex 2
4 1 4 3 4 0 2
B 0 1 3 1 B T
1 1 7
2 7 5 2 4 3 5
(3 4) 3 1 2
(4 3)
28. Matrix Equality
• Two matrices are equal iff (if and only if) all of
their elements are identical
• Note: your data set is a matrix.
29. Matrix Equality
Ex1. Assume A = B find x , y ,z
1 2 x 2
A 3 0 ,B 3 y
1 4 z 4
Solution. If A = B that mean
x = 1
y = 0
z = -1
30. Matrix Equality
Ex2. Assume C = D find x , y ,z
x y 1 4 3 4 1 4 3
C 0 1 3 1 ,D 0 1 3 1
2 7 5 2 y 7 5 z
Solution. If C = D that mean y = 2 , z = 2 and
x + y = 4 thus x + 2 = 4
then x = 2
32. Matrix Addition
A new matrix C may be defined as the
additive combination of matrices A and
B where: C = A + B
is defined by:
Cij Aij Bij
Note: all three matrices are of the same dimension
33. Addition
a11 a12
A
If a 21 a 22
b11 b12
and B
b 21 b 22
a11 b11 a12 b12
then C
a 21 b 21 a 22 b22
34. If A and B are both m n matrices then the sum of A
and B, denoted A + B, is a matrix obtained by adding
corresponding elements of A and B.
corresponding elements of A and B.
add
addadd add
add
these
add
these these 0 4
3
111 222 222these
theseB
these 33 300 0 44 4
AA 1
A 0 1 22 22
111 333
B B 2 33 00 444
B 1
B 2 21 1 4 4
A 00
A 0 B 2 1 4
11 33 2 1 4
0 2 1 4
2 2
22 22 6
A
AA B
BB 22 2
2 2 6
66
A
AA B
BB 2 0 1
22 0
38. Matrix Subtraction
C = A - B
Is defined by
Cij Aij Bij
Note: all three matrices are of the same dimension
39. Subtraction
a11 a12
A
If a 21 a 22
b11 b12
and B
b 21 b 22
then
a11 b11 a12 b12
C
a 21 b21 a 22 b22
40. Addition and Subtraction (cont.)
a11 b11 c11
• Where a12 b12 c12
a 21 b21 c 21
a 22 b22 c 22
a31 b31 c31
a32 b32 c32
• Hence 1 2 4 6 3 4
3 4 4 6 1 2
5 6 4 6 1 0
41. Operations with Matrices (Scalar Multiple)
Scalar Multiple
If A is a matrix and r is a number (sometimes
called a scalar in this context), then the scalar
multiple, rA, is obtained by multiplying every
entry in A by r. In symbols, (rA)ij = raij .
Example:
3 4 1 6 8 2
2
6 7 0 12 14 0
41
42. Scalar Multiplication
• To multiply a scalar times a matrix, simply
multiply each element of the matrix by the
scalar quantity
a11 a12 2a11 2a12
2
a21 a22 2a21 2a22
43. If A is an m n matrix and s is a scalar, then we let kA
denote the matrix obtained by multiplying every
element of A by k. This procedure is called scalar
multiplication.
1 2 2 31 3 2 3 2 3 6 6
A 3A
0 1 3 30 3 1 33 0 3 9
PROPERTIES OF SCALAR MULTIPLICATION
k hA kh A
k h A kA hA
k A B kA kB
44. The m n zero matrix, denoted 0, is the m n
matrix whose elements are all zeros.
0 0
0 0 0
0 0
1 3
2 2
A 0 A
A ( A) 0
0 A 0
45. Operations with Matrices (Product)
Product
If A has dimensions k × m and B has dimensions m × n, then the product
AB is defined, and has dimensions k × n. The entry (AB)ij is obtained
by multiplying row i of A by column j of B, which is done by multiplying
corresponding entries together and then adding the results i.e.,
b1 j
b2 j
( ai1 ai 2 ... aim ) ai1b1 j ai 2b2 j ... aim bmj .
bmj
Example
a b aA bC aB bD
A B
c d . cA dC cB dD
C D
e f eA fC eB fD
1 0 0
0 1 0
Identity matrix I for any m n matrix A, AI A and for
0 0 1 n n
any n m matrix B, IB B.
46. Matrix Multiplication (cont.)
• To multiply a matrix times a matrix, we write
• A times B as AB
• This is pre-multiplying B by A, or post-
multiplying A by B.
47. Matrix Multiplication (cont.)
• In order to multiply matrices, they must be
conformable (the number of columns in A
must equal the number of rows in B.)
• an (mxn) x (nxp) = (mxp)
• an (mxn) x (pxn) = cannot be done
• a (1xn) x (nx1) = a scalar (1x1)
48. Matrix Multiplication (cont.)
• The general rule for
matrix multiplication
is:
N
cij aik bkj where i 1,2,..., M , and j 1,2,..., P
k 1
52. Computation: A x B = C
a11 a12
A [2 x 2]
a 21 a 22
b11 b12 b13
B [2 x 3]
b 21 b 22 b 23
a11b11 a12b21 a11b12 a12b22 a11b13 a12b23
C
a 21b11 a 22b21 a 21b12 a 22b22 a 21b13 a 22b23
[2 x 3]
53. Computation: A x B = C
2 3
111
A 11 and B
1 0 2
1 0
[3 x 2] [2 x 3]
A and B can be multiplied
2 *1 3 *1 5 2 *1 3 * 0 2 2 *1 3 * 2 8 528
C 1*1 1*1 2 1*1 1* 0 1 1*1 1* 2 3 213
1*1 0 *1 1 1*1 0 * 0 1 1*1 0 * 2 1 111
[3 x 3]
54. Computation: A x B = C
2 3
111
A 11 and B
1 0 2
1 0
[3 x 2] [2 x 3]
Result is 3 x 3
2 *1 3 *1 5 2 *1 3 * 0 2 2 *1 3 * 2 8 528
C 1*1 1*1 2 1*1 1* 0 1 1*1 1* 2 3 213
1*1 0 *1 1 1*1 0 * 0 1 1*1 0 * 2 1 111
[3 x 3]
55. The multiplication of matrices is easier shown
than put into words. You multiply the rows of
the first matrix with the columns of the second
adding products Find AB
2 4
3 2 1
A B 1 3
0 4 1
3 1
3 22
3 22 11 1 3 5
First we multiply across the first row and down the
first column adding products. We put the answer in
the first row, first column of the answer.
56. Find AB
2 4
3 2 1
A B 1 3
0 4 1
3 1
5 77
5 7
AB
AB 0034 4 442313 3 1 1 3 11 1
0 24 4 3
3 4
0 2 2 1 1 7
1
1 11
Notice the sizes of A and B and the size of the product AB.
Now we multiplyacross first first androw and down
Now we multiply across the row rowrow and down
We multiplied across the second down first the
second and down
the second column we’ll put the the answerthe the
second column andand we’ll put answer inrow, first
the first column and the answer inanswer in the
column so we put we’ll put the the first in
second row, first column.
row, second second column.
second row, column.
first column.
57. To multiply matrices A and B
look at their dimensions
m n n p
MUST BE
SAME
SIZE OF
PRODUCT
If the number of columns of A does not
equal the number of rows of B then the
product AB is undefined.
58. Now let’s look at the product BA.
2 4 3 2 1
A
B 1 3 0 4 1
3 1 3221 2 2 3 14340 4 102
1111 3 21 3 404 943
323 1 011 612
4 14
3 2 2 3
across third row 66 12
6 12
6 12
12 222
2
as we first row
acrossgo down
across first row
third row
second
third asgodown
as wecoluacross
row we go
as wego down BA
BA
BA
BA 33 14
BA 3 14 14 44
4
first column: as
secondrow
secondfirst
first second
downcolumn:
thirdcolumn:
column:
we go down
column:
column: 9 10
9 10 4
Commuter's Beware!
third column:
mn:
Completely different than AB! AB BA
59. PROPERTIES OF MATRIX
MULTIPLICATION
A BC AB C
A B C AB AC
A B C AC BC
AB BA
Is it possible for AB = BA ? ,yes it is possible.
60. 2 1 2 Multiplying a
What is AI? 0 1 5 A matrix by the
1 20 3 0
2 identity gives
the matrix back
I3 2 011 0
2 again.
What is IA? 0 1 5 A
1 0 0
2 1 2 2 020 1
3
A 0 1 5
I3 0 1 0
2 2 3 0 0 1
an n n matrix with ones on the main
diagonal and zeros elsewhere
61. Matrix multiplication is not
Commutative
• AB does not necessarily equal BA
• (BA may even be an impossible operation)
63. Laws of Matrix Algebra
• The matrix addition, subtraction, scalar multiplication and matrix
multiplication, have the following properties.
Associative Laws :
A (B C) (A B) C, (AB)C A(BC).
Commutative Law for Addition :
A B B A
Distributive Laws :
A(B C) AB AC, (A B)C AC BC.
64. An example - cont
• Since the matrix product is a scalar found by
summing the elements of the vector squared.
65. Determinants
• Determinant is a scalar
– Defined for a square matrix
– Is the sum of selected products of the elements of the matrix, each product
being multiplied by +1 or -1
a11 a12 a1n
a21 a22 a2 n n
i j
n
det( A) aij ( 1) M ij aij ( 1)i j M ij
j 1 i 1
an1 an 2 ann
• Mij=det(Aij), Aij is the (n-1) (n-1)
submatrix obtained by deleting row
i and column j from A.
66. Determinants
a b
• The determinant of a 2 ×2 matrix A is det( A) ad bc
c d
• The determinant of a 3 ×3 matrix is
a11 a12 a13
11
a22 a23 1 2
a21 a23 13
a21 a22
a21 a22 a23 a11 ( 1) a12 ( 1) a13 ( 1)
a32 a33 a31 a33 a31 a32
a31 a32 a33
Example
1 2 3
5 6 4 6 4 5
4 5 6 1( 1)1 1 2( 1)1 2 3( 1)1 3
8 10 7 10 7 8
7 8 10
50 48 2(40 42) 3(32 35) 3
• In Matlab: det(A) = det(A)
67. The Determinant of a Matrix
• The determinant of a matrix A is denoted by
|A|.
• Determinants exist only for square matrices.
• They are a matrix characteristic, and they are
also difficult to compute
68. The Determinant for a 2x2 matrix
• If A =
a11 a12
a21 a22
• Then
A a11a22 a12a21
• That one is easy
69. The Determinant for a 3x3 matrix
• If A =
a11 a12 a13
a21 a22 a23
a31 a32 a33
• Then
A a11a22a33 a11a23a32 a12a23a31 a12a21a33 a13a21a32 a13a22a31
70. Determinants
• For 4 x 4 and up don't try. For those
interested, expansion by minors and cofactors
is the preferred method.
• (However the spaghetti method works well!
Simply duplicate all but the last column of the
matrix next to the original and sum the
products of the diagonals along the following
pattern.)
71. Properties of Determinates
• Determinants have several mathematical properties
which are useful in matrix manipulations.
– 1 |A|=|A'|.
– 2. If a row of A = 0, then |A|= 0.
– 3. If every value in a row is multiplied by k, then |A| =
k|A|.
– 4. If two rows (or columns) are interchanged the
sign, but not value, of |A| changes.
– 5. If two rows are identical, |A| = 0.
72. Properties of Determinates
– 6. |A| remains unchanged if each element of
a row or each element multiplied by a constant, is
added to any other row.
– 7. Det of product = product of Det's |AB| =
|A| |B|
– 8. Det of a diagonal matrix = product of the
diagonal elements
73. Matrix Division
We have seen that for 2x2 (“two by two”)
matrices A and B then AB BA
To divide matrices we need to define
what we mean by division!
With numbers or algebra we use b/a to
solve ax=b. The equivalent in 2x2
matrices is to solve AX=B where A, B
and X are 2x2 matrices.
74. Inverse Matrix
In numbers, the inverse of 3 is 1/3 = 3-1
In algebra, the inverse of a is 1/a = a-1
In matrices, the inverse of A is A-1
3-1 is defined so that 3 x 3-1 = 3-1 x 3 = 1
a-1 is defined so that a x a-1 = a-1 x a = 1
A-1 is defined so that A A-1 = A-1 A = I
However, for a square matrix A there is
not always an inverse A-1
75. Inverse Matrix
In matrices, the inverse of A is A-1
A-1 is defined so that A A-1 = A-1 A = I
However, for a square matrix A there is
not always an inverse A-1
If A-1 does not exist then the matrix is
said to be singular
If A-1 does exist then the matrix is said to
be non-singular
76. Inverse Matrix
In matrices, the inverse of A is A-1
A-1 is defined so that A A-1 = A-1 A = I
A square matrix A has an inverse if, and
only if, A is non-singular.
If A-1 does exist the the solution to AX=B
is
X = A-1 B
77. Inverse Matrix
A-1 is defined so that A A-1 = A-1 A = I
If A-1 does exist the the solution to AX=B is
AX = B
Pre-multiply by A-1 A-1AX = A-1B
But A-1A = I so IX = A-1B
X = A-1B
78. Inverse Matrix
AX = B
Pre-multiply by A-1 A-1AX = A-1B
But A-1A = I so IX = A-1B
X = A-1B
If the inverse of A is A-1 then the inverse of A-1
is A. This is because if AC = I then CA = I, and
also any matrix inverse is unique.
79. Inverse Matrix
If the inverse of A is A-1 then the inverse
of A-1 is A. This is because if AC = I then
CA = I, and also any matrix inverse is
unique. 2 1
B
0 3
What is the inverse of
1 u v
let B 1 1 3 1
w x B
6 0 2
Then solve for u, v, w, x
80. General Inverse Matrix
a b 1 u v 1 d b
C let C
c d w x D c a
where D ad bc
au bw 1 c acu bcw c
cu dw 0 a cau daw 0
av bx 0 Subtract :
cv dx 1 (ad bc) w c
81. Inverse of a Matrix
• Definition. If A is a square matrix, i.e., A has dimensions n×n. Matrix A is
nonsingular or invertible if there exists a matrix B such that AB=BA=In. For
example.
2 1 2 1 1 1
1 1 3 3 3 3 3 3 1 0
1 2 1 1 2 2 1 2 0 1
3 3 3 3 3 3
Common notation for the inverse of a matrix A is A-1
The inverse matrix A-1 is unique when it exists.
If A is invertible, A-1 is also invertible A is the inverse matrix of A-1.
(A-1)-1=A.
• Matrix division:
If A is an invertible matrix, then (AT)-1 = (A-1)T A/B = AB-1
• In Matlab: A-1 = inv(A)
82. Calculation of Inversion using Determinants
Def: For any n n matrix A, let Cij denote the (i,j)th
cofactor of A, that is, (-1)i+j times the determinant of
the submatrix obtained by deleting row i and
column j form A, i.e., Cij = (-1)i+j Mij . The n n matrix
whose (i,j)th entry is Cji, the (j,i)th cofactor of A is
called the adjoint of A and is written adj A.
Thm: Let A be a nonsingular matrix. Then,
1
A-1 adj A.
det A thus
83. Calculation of Inversion using Determinants
2 4 5
Example: find the inverse of the matrix
A 0 3 0
Solve: 1 0 1
3 0 0 0 0 3
C11 3, C12 0, C13 3,
0 1 1 1 1 0
4 5 2 5 2 4
C21 4, C22 3, C23 4,
0 1 1 1 1 0
4 5 2 5 2 4
C31 15, C32 0, C33 6,
3 0 0 0 0 3
det A 9,
C11 C21 C31 3 4 15
adjA C12 C22 C32 0 3 0 .
C13 C23 C33 3 4 6
3 4 15
So, A 1 1
0 3 0 .
thus Determinants to find the
Using
9 inverse of a matrix can be very
3 4 6
complicated. Gaussian elimination is
more efficient for high dimension
matrix.
84. Calculation of Inversion using Gaussian Elimination
Elementary row operations:
o Interchange two rows of a matrix
o Change a row by adding to it a multiple of
another row
o Multiply each element in a row by the same
nonzero number
• To calculate the inverse of matrix A, we apply the elementary row
operations on the augmented matrix [A I] and reduce this matrix to the
form of [I B]
• The right half of this augmented matrix B is the inverse of A
85. Calculation of inversion using Gaussian elimination
a11 , , a1n a11 , , a1n 1 0 0
a21 , , a2 n a21 , , a2 n 0 1 0
A [A I]
an1 , , ann
an1 , , ann 0 0 1
I is the identity matrix, and use Gaussian elimination to
obtain a matrix of the form 1 0 0 b11 b12 b1n
0 1 0 b21 b22 b2 n
0 0 1 bn1 bn 2 bnn
The matrix b11 b12 b1n
b21 b22 b2 n
B is then the matrix inverse of A
bn1 bn 2 bnn
87. Can we find a matrix to multiply the first
matrix by to get the identity?
1
3 1 1 1 0
? 32
4 2 2 0 1
2
1
1 3 1 1 0
2
3 4 2 0 1
2
2
Let A be an n n matrix. If there exists a matrix B such
that AB = BA = I then we call this matrix the inverse of
A and denote it A-1.
88. If A has an inverse we say that A is nonsingular.
If A-1 does not exist we say A is singular.
To find the inverse of a matrix we put the matrix A, a
To find the inverse of a matrix we put the matrix A, a
line and then the identity matrix. We then perform row
line and then the identity matrix. We then perform row
operations on matrix A to turn it into the identity. We
operations on matrix A to turn it into the identity. We
carry the row operations across and the right hand side
carry the row operations across and the right hand side
will turn into the inverse.
inverse. 1 3
A
2 7
1 3 1 0
1 3 1 0
2 7 0 1 r2 0 1 2 1
1 3 1 0 r 1 r2 1 0 7 3
2r1+r 0 1 2 1 0 1 2 1
89. 1 3 7 3
A A 1
2 7 2 1
Check this answer by multiplying. We
should get the identity matrix if we’ve
found the inverse.
1 1 0
AA
0 1
91. We can use A-1 to solve a system of equations
x 3y 1 To see how, we can re-write a
system of equations as
2x 5 y 3 matrices.
Ax b
coefficien variable constant
t matrix matrix matrix
1 3 x 1
2 5 y 3
92. Ax b left multiply both
sides by the
inverse of A
1 1
A Ax A b
This is just the identity
but the identity
1 times a matrix just
Ix A b gives us back the
matrix so we have:
This then gives us a formula
1
for finding the variable
matrix: Multiply A inverse x A b
by the constants.
93. x 3y 1 1 3 find the
A
2x 5 y 3 2 5 inverse
1 3 1 0 1 3 1 0
2 5 0 1 - 0 1 2 1
2r1+r2
1 3 1 0 r 1- 1 0 5 3
3r2
- 0 1 2 1 0 1 2 1
r2
1 5 3 1 4 x This is the
A b answer to
2 1 3 1 y the system
94. Systems of Equations in Matrix Form
The system of linear equations a11 x1 a12 x2 a13 x3 a1n xn b1
a21 x1 a22 x2 a23 x3 a2 n xn b2
ak1 x1 ak 2 x2 ak 3 x3 akn xn bk
can be rewritten as the matrix equation Ax=b, where
x1 b1
a11 a1n
x2 b2
A , x , b .
ak1 akn
xn bk
If an n n matrix A is invertible, then it is nonsingular, and the unique
solution to the system of linear equations Ax=b is x=A-1b.
95. Example: solve the linear system 4x y 2z 4
5x 2 y z 4
x 3z 3
Matrix Inversion
AX b
d
4 1 2 x 4
A 5 2 1 ;X y ;b 4
1 0 3 z 3
X A 1b
6 -3 -3
1
A -1 -14 10 6
6
-2 1 3
x 6 -3 -3 4
1
y -14 10 6 4
6
z -2 1 3 3
x 1 2; y 1 3; z 5 6
96. Matrix Inversion
1 1
B B BB I
Like a reciprocal Like the number one
in scalar math in scalar math
97. Linear System of Simultaneous Equations
First precinct: 6 arrests last week equally divided
between felonies and misdemeanors.
Second precinct: 9 arrests - there were twice as
many felonies as the first precinct.
1st Precinct : x1 x2 6
2nd Pr ecinct : 2x1 x2 9
98. Solution Note:
11
21 i
11
2 1
Inverse of s
11 x1 6
*
21 x2 9
11 11 x1 11 6 Premultiply both sides
* * *
2 1 21 x2 2 1 9 by inverse matrix
10 x1 3 A square matrix
* multiplied by its inverse
01 x2 3
results in the identity
matrix.
x1 3 A 2x2 identity matrix
x2 3 multiplied by the 2x1
matrix results in the
original 2x1 matrix.
99. General Form
n equations in n variables:
n
aijxj bi or Ax b
j 1
unknown values of x can be found using the
inverse of matrix A such that
1 1
x A Ax A b
