If the inverse of the matrix A  =\(\begin{bmatrix}3 & 1 & 2 \\ 4&2 & 1\\ 2 & a & 1 \end{bmatrix}\)does not exist then the value of a is

Concept:

Consider a matrix A and let its inverse be A^-1

\(\rm {A^{ - 1}} = \frac{{{\rm{adj\;}}\left( {\rm{A}} \right){\rm{\;}}}}{{{\rm{det\;}}\left( {\rm{A}} \right)}}\)

Here; adj (A) is adjoint of matrix A and det (A) is determinant of matrix A.

⇒ If det (A) ≠ 0, so the inverse of a matrix exists.

⇒ If det (A) = 0, so inverse of a matrix does not exist.

Calculation:

Given A = \(\begin{bmatrix}3 & 1 & 2 \\ 4&2 & 1\\ 2 & a & 1 \end{bmatrix}\)

For A^-1 does not exist the |A| = 0

|A| = \(\begin{vmatrix}3 & 1 & 2 \\ 4&2 & 1\\ 2 & a & 1 \end{vmatrix}\) = 0

|A| = 3(2 - a) - 1(4 - 2) + 2(4a - 4)

|A| = 6 - 3a - 2 + 8a - 8

|A| = 5a - 4

|A| = 0

5a - 4 = 0

∴ a = \(\frac 4 5\)