Engineering Mathematics

Trace of Matrix Product

Consider the following two statements with respect to the matrices \(A_{m\times n}\), \(B_{n\times m}\), \(C_{n\times n}\) and \(D_{n\times n}\).

Which one of the following holds?

Singular Matrix

Which one of the following matrices is singular?

Determinant of Inverse Matrix

If \(A= \begin{bmatrix} 1 & 2 & 3\\ 0 & 1 & 2\\ 0 & 0 & 4 \end{bmatrix}\), then \(\det(A^{-1})\) is:

Trace and Determinant of Matrix

The matrix \(A= \begin{bmatrix} a & 0 & 3 & 7\\ 2 & 5 & 1 & 3\\ 0 & 0 & 2 & 4\\ 0 & 0 & 0 & b \end{bmatrix}\) has \(\det(A)=100\) and \(\operatorname{tr}(A)=14\).

Find \(|a-b|\).

Determinant of Matrices

Let \(A= \begin{bmatrix} 1&2&3&4\\ 4&1&2&3\\ 3&4&1&2\\ 2&3&4&1 \end{bmatrix}\) and \(B= \begin{bmatrix} 3&4&1&2\\ 4&1&2&3\\ 1&2&3&4\\ 2&3&4&1 \end{bmatrix}\).

Which one of the following is true?