Question

If A is a diagonal matrix of order is commutative with every square matrix of order under multiplication and trace (A) = 9, then |A| =​

Answer 1

Given: A is a diagonal matrix of order 3x3, trace (A) = 9

To find: |A| = ?

Solution:

• As we have given that the matrix is diagonal matrix, it means that the matrix has a non zero diagonal and rest other elements are 0.

• So the matrix will be:

                $\left[\begin{array}{ccc}x&0&0\\0&x&0\\0&0&x\end{array}\right]$

• Here we have taken all the diagonal elements same because it is given that the matrix is commutative with every square matrix.

• Trace of a matrix is sum of the diagonal elements.

• As the trace is given as 9, this concludes:

               x+x+x=9

               3x=9

               x=3

• Putting x=3 in the matrix we get:

               $\left[\begin{array}{ccc}3&0&0\\0&3&0\\0&0&3\end{array}\right]$

• Solving this matrix as determinant, we get:

             = 3(3x3)

             = 27

Answer:

              So, determinant of (A) is 27