# Properties of the Identity Matrix

A matrix multiplied by its inverse is equal to the identity matrix.

*A · **A*^{-1} = A^{-1 }· *A = I*

All the elements of the matrix apart from the diagonal are zero.

For an m × n matrix *A*:

*I*_{m }A = A I_{n} = A

**Example 1: **If , then find *M × I*, where *I* is an identity matrix.

**Solution: **

As *M* is square matrix of order 2×2, the identity matrix *I* needs to be of the same order 2×2. [Rule for Matrix Multiplication.]

Thus:

**Example 2: **Determine the value of x, y and z if:

**Solution: **

On the L.H.S we have:

Now, we compare the L.H.S matrix with the R.H.S matrix in order to determine the values of x, y and z.

y = 4x ……….(1)

z – 1 = 6 ……(2)

x – 2 = y/2 ……(3)

From (2) we can determine the value of z, which is z = 7.

From (3) we can simplify the equation to y = 2x – 4

We now have two equations with two unknown factors, namely x and y:

y = 2x – 4

y = 4x

To determine the value of x and y, simply insert y = 4x into y = 2x – 4, which will give us:

4x = 2x – 4 => x = – 2

And y = 4x = – 8

Hence (*x*, *y*, *z*) = (*-2*, *-8*, *7*)