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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
IA = A In = 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)

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