If A is the identity matrix of order 3 and B is its transpose, then what is the value of the determinant of the matrix C = A + B?

Concept:

1). Transpose of a Matrix:

     The new matrix obtained by interchanging the rows and columns of the original matrix is called

     the transpose of the matrix.

     For example: \(A = \begin{bmatrix} a &b & c \\[0.3em] x & y & z \\[0.3em] \end{bmatrix}\)⇒ \(A^T = \begin{bmatrix} a &x \\[0.3em] b & y \\[0.3em] c & z \\[0.3em] \end{bmatrix}\)

     It is denoted by A' or A^T.
2). For the addition and subtraction of two matrices, the order of matrices should be equal.

Calculation:

Given:

\(A = \begin{bmatrix} 1 &0 & 0 \\[0.3em] 0& 1 &0\\[0.3em] 0 & 0 & 1 \end{bmatrix}\)and \(B = A^T= \begin{bmatrix} 1 &0 & 0 \\[0.3em] 0& 1 &0\\[0.3em] 0 & 0 & 1 \end{bmatrix}\)

C = A + B

⇒ \(C = \begin{bmatrix} 1 &0 & 0 \\[0.3em] 0& 1 &0\\[0.3em] 0 & 0 & 1 \end{bmatrix}+ \begin{bmatrix} 1 &0 & 0 \\[0.3em] 0& 1 &0\\[0.3em] 0 & 0 & 1 \end{bmatrix}\)

⇒ \(C = \begin{bmatrix} 2 &0 & 0 \\[0.3em] 0& 2 &0\\[0.3em] 0 & 0 & 2 \end{bmatrix}\)

⇒ ∣C∣ = 2(4 - 0)

⇒ ∣C∣ = 8

∴ The determinant of matrix C is 8.