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Concept

Multiplicative Inverse of a Matrix

Let

A

and

B

be two square matrices with the same dimensions. If the result of multiplying

A

B

from either side is the identity matrix

I,

then

B

is the multiplicative inverse of

A .

A B = B A = I

The multiplicative inverse of a matrix $A$ is usually denoted $A^{- 1} .$

Example

Consider two

3 \times 3

matrices.

A = ⎣ ⎢ ⎡ 1 - 2 - 1 011212 ⎦ ⎥ ⎤ B = ⎣ ⎢ ⎡ - 1 - 3 1 - 2 - 4 1 25 - 1 ⎦ ⎥ ⎤

It can be shown that

B

is the multiplicative inverse of

A

by calculating the products

A \cdot B

and

B \cdot A .

First, the product

A \cdot B

is calculated.

⎣ ⎢ ⎡ 1 - 2 - 1 011212 ⎦ ⎥ ⎤ \cdot ⎣ ⎢ ⎡ - 1 - 3 1 - 2 - 4 1 25 - 1 ⎦ ⎥ ⎤

▼

Multiply matrices

MultMatrices

Multiply matrices

⎣ ⎢ ⎡ 1 (- 1) + 0 (- 3) + 2 (1) - 2 (- 1) + 1 (- 3) + 1 (1) - 1 (- 1) + 1 (- 3) + 2 (1) 1 (- 2) + 0 (- 4) + 2 (1) - 2 (- 2) + 1 (- 4) + 1 (1) - 1 (- 2) + 1 (- 4) + 2 (1) 1 (2) + 0 (5) + 2 (- 1) - 2 (2) + 1 (5) + 1 (- 1) - 1 (2) + 1 (5) + 2 (- 1) ⎦ ⎥ ⎤

Multiply

⎣ ⎢ ⎡ - 1 + 0 + 2 2 + (- 3) + 1 1 + (- 3) + 2 - 2 + 0 + 2 4 + (- 4) + 1 2 + (- 4) + 2 2 + 0 + (- 2) - 4 + 5 + (- 1) - 2 + 5 + (- 2) ⎦ ⎥ ⎤

AddNeg

$a + (- b) = a - b$

⎣ ⎢ ⎡ - 1 + 0 + 2 2 - 3 + 1 1 - 3 + 2 - 2 + 0 + 2 4 - 4 + 1 2 - 4 + 2 2 + 0 - 2 - 4 + 5 - 1 - 2 + 5 - 2 ⎦ ⎥ ⎤

AddSubTerms

Add and subtract terms

⎣ ⎢ ⎡ 100010001 ⎦ ⎥ ⎤

Next, the product

B \cdot A

is calculated.

⎣ ⎢ ⎡ - 1 - 3 1 - 2 - 4 1 25 - 1 ⎦ ⎥ ⎤ \cdot ⎣ ⎢ ⎡ 1 - 2 - 1 011212 ⎦ ⎥ ⎤

▼

Multiply matrices

MultMatrices

Multiply matrices

⎣ ⎢ ⎡ - 1 (1) + (- 2) (- 2) + 2 (- 1) - 3 (1) + (- 4) (- 2) + 5 (- 1) 1 (1) + 1 (- 2) + (- 1) (- 1) - 1 (0) + (- 2) (1) + 2 (1) - 3 (0) + (- 4) (1) + 5 (1) 1 (0) + 1 (1) + (- 1) (1) - 1 (2) + (- 2) (1) + 2 (2) - 3 (2) + (- 4) (1) + 5 (2) 1 (2) + 1 (1) + (- 1) (2) ⎦ ⎥ ⎤

Multiply

⎣ ⎢ ⎡ - 1 + 4 + (- 2) - 3 + 8 + (- 5) 1 + (- 2) + 1 0 + (- 2) + 2 0 + (- 4) + 5 0 + 1 + (- 1) - 2 + (- 2) + 4 - 6 + (- 4) + 10 2 + 1 + (- 2) ⎦ ⎥ ⎤

AddNeg

$a + (- b) = a - b$

⎣ ⎢ ⎡ - 1 + 4 - 2 - 3 + 8 - 5 1 - 2 + 1 0 - 2 + 2 0 - 4 + 5 0 + 1 - 1 - 2 - 2 + 4 - 6 - 4 + 10 2 + 1 - 2 ⎦ ⎥ ⎤

AddSubTerms

Add and subtract terms

⎣ ⎢ ⎡ 100010001 ⎦ ⎥ ⎤

Since

A \cdot B = I

and

B \cdot A = I,

B

is the multiplicative inverse of

A

and

B = A^{- 1} .

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Example