Finding the inverse of a matrix Using Gauss - Jordan Elimination

1. INVERSE
OF A
MATRIX

2. FIND THE PRODUCT
𝐴 =
−1 2
−1 1
𝐵 =
1 −2
1 −1 𝐶 =
1 2
1 1
𝐴𝐵 =
1 0
0 1
𝐵𝐴 =
1 0
0 1
𝐴𝐶 =
1 0
0 −1
𝐶𝐴 =
−3 4
−2 3

3. DEFINITION:
An 𝑛 × 𝑛 matrix A is invertible (nonsingular) when there
exists an 𝑛 × 𝑛 matrix B such that,
𝑨𝑩 = 𝑩𝑨 = 𝑰 𝒏
where 𝑰 𝒏 is the identity matrix of order n. The matrix B is called
the (multiplicative) inverse of A.
A matrix that does not have an inverse is called
noninvertible (singular).
𝑨𝑪 ≠ 𝑪𝑨 ≠ 𝑰 𝒏

4. FINDING THE INVERSE OF A MATRIX
𝐴 =
1 4
−1 −3
Solution:
1 4
−1 −3
𝑥11 𝑥12
𝑥21 𝑥22
=
1 0
0 1
𝐴𝑋 = 𝐼 𝑛
𝑥11 + 4𝑥21 𝑥12 + 4𝑥22
−𝑥11 − 3𝑥21 −𝑥12 − 3𝑥22
=
1 0
0 1
𝑥11 + 4𝑥21 = 1
−𝑥11 − 3𝑥21 = 0
𝑥12 + 4𝑥22 = 0
−𝑥12 − 3𝑥22 = 1
𝑋 = 𝐴−1
=
−3 −4
1 1

5. FINDING THE INVERSE OF A MATRIX
1 4 1
−1 −3 0
1 4 1
−1 −3 0
1
−1
4 1 0
−3 0 1
𝑈𝑠𝑖𝑛𝑔 𝐺𝐴𝑈𝑆𝑆 − 𝐽𝑂𝑅𝐷𝐴𝑁 𝐸𝑙𝑖𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛
1
0
4 1 0
1 1 1
𝑅2 + 𝑅1 → 𝑅2
1
0
0 −3 −4
1 1 1
𝑅1 + (−4)𝑅2→ 𝑅1
1
−1
4 1 0
−3 0 1
1
0
0 −3 −4
1 1 1
→
𝐴 𝐼 𝐼 𝐴

6. Let A be a square matrix of order n.
1. Write the 𝑛 × 2𝑛 matrix that consists of the given matrix A on
the left and the 𝑛 × 𝑛 identity matrix I on the right to obtain
𝐴 𝐼 . This matrix is called adjoining matrix I to matrix A.
2. If possible, row reduce A to I using elementary row operations
on the entire matrix 𝐴 𝐼 . The result will be the matrix
𝐼 𝐴−1 . If this is not possible, then A is noninvertible
(nonsingular).
3. 𝐴𝐴−1 = 𝐼 = 𝐴−1 𝐴
FINDING THE INVERSE OF A MATRIXGAUSS – JORDAN ELIMINATION

7. Find the inverse of the matrix.
𝐴 =
1 −1 0
1 0 −1
−6 2 3
FINDING THE INVERSE OF A MATRIXGAUSS – JORDAN ELIMINATION