This document contains 4 sections that discuss properties of matrices: 1) It defines zero matrices and shows that the determinant of a zero matrix is 0. 2) It proves that the transpose of a matrix P is equal to the transpose of its transpose (PT = (PT)T). 3) It solves a system of 4 equations with 4 unknowns to find the values of w, x, y, and z. 4) It provides an expression for the inverse of a non-singular square matrix A in terms of the matrix's cofactors.

1. 2
1. (5 ! )
# $%
1. (Zero matrices) 0 1 2 3 2 3 4 5 4 0 467 2 467 4 5
78 4
2. $ !4 % 0 87 : 7#
; !<! 87 3% :% =:% 24% %
:$ 8 7#
0 0 0
a 21 a 22 a2n
:24% A = 0 :% 4 :@ !
a n1 a n 2 a nn
@ A : (cofactor) ! %
det(A) = a11C11(A) + a12C12(A) + a13C13(A) + G + a1nC1n(A)
= 0+0+0+G+0
= 0
! % det(A) = 0 7 0 A : :% Q.E.D.
p11 p12 p13
2. J 4 24% P = p 21 p 22 p 23 6! P 3% PT @; $
p 31 p 32 p 33
P P = (PT)T (5 ! )
p11 p 21 p 31
@ J4 P 24% ! % PT = p12 p 22 p 32
p13 p 23 p 33
p11 p12 p13
; !<! 87 (PT)T = p 21 p 22 p 23 = P Q.E.D.
p 31 p 32 p 33

2. 3. 21% M$ % !## : 8 (10 ! )
wOx+yOz = O4
4w O x + 3y + z = O8
2w + x + y O z = 0
3w + 2x + y O 3z = 1
$ !## J 4 24% 2 7 8
1 1 1 1 w 4
4 1 3 1 x 8
=
2 1 1 1 y 0
3 2 1 3 z 1
1 1 1 1 4 1 1 1
4 1 3 1 8 1 3 1
det(A) = = O 4, det(A1) = =8
2 1 1 1 0 1 1 1
3 2 1 3 1 2 1 3
1 4 1 1 1 1 4 1
4 8 3 1 4 1 8 1
det(A2) = = O 12, det(A3) = = O4
2 0 1 1 2 1 0 1
3 1 1 3 3 2 1 3
1 1 1 4
4 1 3 8
det(A4) = =0
2 1 1 0
3 2 1 1
! % w = 84 = O 2, x = 12 = 3, y = 4 = 1, z = 0
4 4

3. a11 a12 a13
4. J 4 24% A = a 21 a 22 a 23 3% A
a 31 a 32 a 33
V4 AO 1 (10 ! )
: AO 1 = det(A ))
adj(
A
T
C11 ( A ) C12 ( A ) C13 ( A )
1
= det( A ) C 21 ( A ) C 22 ( A ) C 23 ( A )
C31 ( A ) C32 ( A ) C33 ( A )
C11 ( A ) C 21 ( A ) C31 ( A )
1
= det( A ) C12 ( A ) C 22 ( A ) C32 ( A )
C13 ( A ) C 23 ( A ) C33 ( A )
+ (a 22 a 33 a 32 a 23 ) ( a12 a 33 a 32 a13 ) + (a12 a 23 a 22a13 )
1
= det( A ) (a 21a 33 a 31a 23 ) + (a11a 33 a 31a13 ) ( a11a 23 a 21a13 )
+ (a 21a 32 a 31a 22 ) (a11a 32 a 31a12 ) + (a11a 22 a 21a12 )