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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 by B from either side is the identity matrix I, then B is the multiplicative inverse of A.

AB=BA=I

The multiplicative inverse of a matrix A is usually denoted A^(- 1).

Example

Consider two 3* 3 matrices. A= 1 & 0 & 2 - 2 & 1 & 1 - 1 & 1 & 2 B= - 1 & - 2 & 2 - 3 & - 4 & 5 1 & 1 & - 1 It can be shown that B is the multiplicative inverse of A by calculating the products A* B and B* A. First, the product A* B is calculated.

1 & 0 & 2 - 2 & 1 & 1 - 1 & 1 & 2 * - 1 & - 2 & 2 - 3 & - 4 & 5 1 & 1 & - 1

▼

Multiply matrices

MultMatrices

Multiply matrices

1(- 1)+0(- 3)+2(1) & 1(- 2)+0(- 4)+2(1) & 1(2)+0(5)+2(- 1) - 2(- 1)+1(- 3)+1(1) & - 2(- 2)+1(- 4)+1(1) & - 2(2)+1(5)+1(- 1) - 1(- 1)+1(- 3)+2(1) & - 1(- 2)+1(- 4)+2(1) & - 1(2)+1(5)+2(- 1)

Multiply

- 1+0+2 & - 2+0+2 & 2+0+(- 2) 2+(- 3)+1 & 4+(- 4)+1 & - 4+5+(- 1) 1+(- 3)+2 & 2+(- 4)+2 & - 2+5+(- 2)

AddNeg

a+(- b)=a-b

- 1+0+2 & - 2+0+2 & 2+0-2 2-3+1 & 4-4+1 & - 4+5-1 1-3+2 & 2-4+2 & - 2+5-2

AddSubTerms

Add and subtract terms

1 & 0 & 0 0 & 1 & 0 0 & 0 & 1

Next, the product B* A is calculated.

- 1 & - 2 & 2 - 3 & - 4 & 5 1 & 1 & - 1 * 1 & 0 & 2 - 2 & 1 & 1 - 1 & 1 & 2

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Multiply matrices

MultMatrices

Multiply matrices

- 1(1)+(- 2)(- 2)+2(- 1) & - 1(0)+(- 2)(1)+2(1) & - 1(2)+(- 2)(1)+2(2) - 3(1)+(- 4)(- 2)+5(- 1) & - 3(0)+(- 4)(1)+5(1) & - 3(2)+(- 4)(1)+5(2) 1(1)+1(- 2)+(- 1)(- 1) & 1(0)+1(1)+(- 1)(1) & 1(2)+1(1)+(- 1)(2)

Multiply

- 1+4+(- 2) & 0+(- 2)+2 & - 2+(- 2)+4 - 3+8+(- 5) & 0+(- 4)+5 & - 6+(- 4)+10 1+(- 2)+1 & 0+1+(- 1) & 2+1+(- 2)

AddNeg

a+(- b)=a-b

- 1+4-2 & 0-2+2 & - 2-2+4 - 3+8-5 & 0-4+5 & - 6-4+10 1-2+1 & 0+1-1 & 2+1-2

AddSubTerms

Add and subtract terms

1 & 0 & 0 0 & 1 & 0 0 & 0 & 1

Since A* B=I and B* A=I, B is the multiplicative inverse of A and B=A^(- 1).

Exercises

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