Inverse Matrices
Matrices: Inverse Definition
•
The inverse of a square matrix A, if it exists, is another matrix B such that
AB=BA=I
For convenience B is usually represented as A -1 . Even though order is
important in matrix multiplication, the inverse matrix is independent of the order
of multiplication. Some properties of inverse A:
1. A -1 exists only if A is nonsingular - that is | A | ≠ 0
Note: | A | is defined as the determinant of A.
2. If A -1 exists then it is unique.
The inverse A -1 is obtained by replacing the elements aij of A with the
cofactor of aij- that is Aij ,transposing the resulting matrix, and finally dividing
by | A| .
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The Identity Matrix In is the n × n matrix for which each main diagonal entry is a
1 and for which all other entries are 0.
⎡1 0 0⎤
⎡1 0 ⎤
I2 = ⎢
I 3 = ⎢⎢0 1 0⎥⎥3 × 3 Identity Matrix
⎥ 2 × 2 Identity matrix
0
1
⎣
⎦
⎣⎢0 0 1⎦⎥
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“Not every square matrix has an inverse. The following rule provides a simple
way for calculating the inverse of a 2 × 2 matrix, when it exists.”
Note: For larger matrices, there is a more general procedure.
Inverse of a 2 × 2 Matrix,
Determinant
⎡a b ⎤
1 ⎡ d − b⎤
.
If A = ⎢
Then, A −1 =
⎥
ad − bc ⎢⎣− c a ⎥⎦
⎣c d ⎦
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Example: Find the inverse of the 2 × 2 Matrix and verify that A A-1 = A-1 A = I2
Let
⎡4 5⎤
A= ⎢ ⎥
⎣2 3⎦
Solution: Apply the formula for finding the inverse of a 2 × 2 Matrix.
Step 1: Find A-1
⎡ 3 − 5⎤
1
4 ⋅ 3 − 5 ⋅ 2 ⎢⎣− 2 4 ⎥⎦
3 − 5⎤
1 ⎡ 3 − 5⎤ ⎡
⎢
= ⎢
=
2
2 ⎥
2 ⎣− 2 4 ⎥⎦ ⎢− 1 2 ⎥
⎣
⎦
-1
-1
Step 2: Verify that A A = A A = I2
5
⎡ 3
⎤
⎡ 3 − 5 ⎤ ⎢4 ⋅ + 5(−1) 4(− ) + 5 ⋅ 2⎥ ⎡1 0⎤
4
5
⎡
⎤
2
2
A A-1= ⎢
⎥=⎢
2 ⎥=⎢ 3
⎥⎢ 2
⎥
2
3
⎢
⎥
⎣
⎦ ⎣− 1 2 ⎦ ⎢ 2 ⋅ + 3(−1) 2(− 5 ) + 3 ⋅ 2⎥ ⎣0 1⎦
2
⎣ 2
⎦
5⎤
5
3
5 ⎤
⎡3
⎡3
− ⎥ ⎡4 5⎤ ⎢ ⋅ 4 + (− ) ⋅ 2
⋅ 5 + (− ) ⋅ 3⎥ ⎡1 0⎤
=
=⎢
A −1 A = ⎢ 2
2
2 ⎢
2
2
2
⎥
⎥
⎢− 1 2 ⎥ ⎣2 3⎦ ⎢ (−1) ⋅ 4 + 2 ⋅ 2 (−1) ⋅ 5 + 2 ⋅ 3 ⎥ ⎣0 1⎦
⎣
⎦
⎣
⎦
A −1 =
•
Inverse of n × n Matrices
“For 3× 3 and larger square matrices, the following technique provides the
most efficient way to calculate their inverses.”
⎡ a11
⎢a
⎢ 21
⎢ M
⎢
⎣ a n1
a12
a 22
M
L a1n
L a2n
O
M
a n 2 L a nn
1 0 L 0⎤
0 1 L 0⎥⎥
M M O M⎥
⎥
0 0 L 1⎦
Note: Use the elementary row operations on this new large matrix to change the
left side into the identity matrix. The right side will be a transformed
automatically into A-1.
⎡ 1 − 2 − 4⎤
Example: Find the inverse of a 3× 3 Matrix. Let A = ⎢⎢ 2 − 3 − 6⎥⎥
⎢⎣− 3 6 15 ⎥⎦
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Solution: Begin with the 3× 6 matrix whose left half is A and whose right half is the
identity matrix.
⎡ 1 − 2 − 4 1 0 0⎤
⎢ 2 − 3 − 6 0 1 0⎥
⎢
⎥
⎢⎣− 3 6 15 0 0 1⎥⎦
Transform the left half of this new matrix into the identity matrix by performing the
following sequence of elementary row operations.
2 R1 + R2 → R2
⎯ −⎯
⎯⎯
⎯→
R3 + R3 → R3
⎯3⎯
⎯⎯
⎯→
⎡1 − 2 − 4 1 0 0 ⎤
⎢0 1
2 − 2 1 0⎥⎥
⎢
⎢⎣0 0
3
3 0 1⎥⎦
⎤
⎡
⎢1 − 2 − 4 1 0 0 ⎥
⎢0 1
2 − 2 1 0⎥
⎢
1⎥
1
1 0
⎥
⎢0 0
3⎦
⎣
1
R3
3
⎯ ⎯→
⎯
2R2 + R1→R1
⎯⎯⎯⎯→
R3 +R2→R2
⎯2⎯
⎯⎯
⎯→
⎤
⎡
⎢1 0 0 − 3 2 0 ⎥
⎢0 1 2 − 2 1 0 ⎥
⎢
1⎥
⎥
⎢0 0 1 1 0
3⎦
⎣
⎡
⎢1 0 0 − 3 2
⎢
⎢0 1 0 − 4 1
⎢
⎢0 0 1 1 0
⎣⎢
⎤
0 ⎥
− 2⎥
⎥
3 ⎥
1 ⎥
3 ⎦⎥
⎡
⎢1 0 0 − 3 2
⎢
Hence, the Inverse Matrix is: A-1 = ⎢0 1 0 − 4 1
⎢
⎢0 0 1 1 0
⎢⎣
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⎤
0 ⎥
− 2⎥
⎥
3 ⎥
1 ⎥
3 ⎥⎦
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