Chapter 3

Find the determinant of the given matrix using cofactor expansion along the first row. \(\left[\begin{array}{cccc}-1 & 0 & 0 & -1 \\ -1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 \\\ 1 & 0 & -1 & -1\end{array}\right]\)

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}0 & 1 & -2 \\ -1 & 0 & 4 \\ 2 & -4 & 0\end{array}\right]\)

Find the determinant of the given matrix using cofactor expansion along the first row. \(\left[\begin{array}{cccc}-5 & 1 & 0 & 0 \\ -3 & -5 & 2 & 5 \\ -2 & 4 & -3 & 4 \\ 5 & 4 & -3 & 3\end{array}\right]\)

Find the determinant of the given matrix using cofactor expansion along the first row. \(\left[\begin{array}{cccc}2 & -1 & 4 & 4 \\ 3 & -3 & 3 & 2 \\ 0 & 4 & -5 & 1 \\\ -2 & -5 & -2 & -5\end{array}\right]\)

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]\)

Let \(A\) be a \(2 \times 2\) matrix; $$A=\left[\begin{array}{ll}a & b \\\c & d\end{array}\right]$$ Show why \(\operatorname{det}(A)=a d-b c\) by computing the cofactor expansion of \(A\) along the first row.