WHAT IS THE RELATIONSHIP BETWEEN THE DETERMINANT OF A SQUARE AND INVERTIBLE MATRIX OF ORDER 3 AND ITS INVERSE? EXPLAIN HOW AND WITH EXAMPLES.

The relationship between these terms is known as , inverse of a matrix.
The inverse of an invertible square matrix A is given by A⁻¹ =
 

Example,
If A =, then find A⁻¹.

Solution:

For the given matrix A, we have |A| = 1(3 − 2) −2 (−9 + 2) + 1 (6 − 2) = 1 + 14 + 4 = 19

Now, A₁₁ = 1, A₁₂ = 7, A₁₃ = 4,

A₂₁ = 8, A₂₂ = −1, A₂₃ = −6,

A₃₁ = 3, A₃₂ = 2, A₃₃ = −7

∴ adj A =

Thus, A⁻¹