1. Multiplicative inverse and solving matrix equations
Matrices
Multiplicative inverse and solving matrix equations

2. Multiplicative inverse
Any number multiplied by its reciprocal equals 1, e.g.
The same occurs in matrices: if
then is called the multiplicative inverse of A.

3. Inverse of a 2 x 2 matrix If A = then A-1 =
The value of is known as the determinant of matrix A and can be written as det A or

4. Example, Ex 16D, Q.1, 4
You do Ex 16D, Q. 4, 5, 8, 9

5. Singular matrix
A matrix that has a determinant of zero, has no inverse and is called a singular matrix.

6. Matrix equations Remember that in an equation such as
4x =9, we need to divide both sides by 4 (or multiply both sides by ¼) to obtain the answer, x = 9/4.
We solve matrix equations the same way:

7. If we want to solve AX = B for X
Pre-multiply both sides by A-1
A-1AX = A-1B
IX = A-1B (A-1A = I)
X = A-1B (IX = X)

8. If we want to solve XA = B for X
Post-multiply both sides by A-1
XAA-1 = BA-1
XI = BA-1 (AA-1 = I)
X = BA-1 (XI = X)

9. If AX = B, then X = A-1B
If XA = B, then X = BA-1

10. Example, Ex 16D, Q.10
You do Ex16D, Q.11