Use row and column operations to compute the determinant of the matrix\(\Mtrx{A}\) below.
\[ A = \left[ \begin{array}{rrrr} 0 & 1 & 3 & 4 \\ 2 & 3 & 1 & 1 \\ 4 & -1 & -1 & 2 \\ 6 & 4 & -1 & 1 \end{array} \right] \]We know that the determinant of an upper triangular matrix is just the product of its diagonal entries. Therefore, we use row and column operations on \(\Mtrx{A}\) to transform \(\Mtrx{A}\) toward triangular shape. At each transformation step we must carefully keep track of the effect of this step on the determinant. So, here goes:
| \(\det (\Mtrx{A})\) | \(=\) | \( \det \left[ \begin{array}{rrrr} {\color{red} 0} & {\color{red} 1} & {\color{red} 3} & {\color{red} 4} \\ {\color{blue} 2} & {\color{blue} 3} & {\color{blue} 1} & {\color{blue} 1} \\ 4 & -1 & -1 & 2 \\ 6 & 4 & -1 & 1 \end{array} \right] \) |
| \(\) | \(= \) | \( - \det \left[ \begin{array}{rrrr} {\color{blue} 2} & {\color{blue} 3} & {\color{blue} 1} & {\color{blue} 1} \\ {\color{red} 0} & {\color{red} 1} & {\color{red} 3} & {\color{red} 4} \\ 4 & -1 & -1 & 2 \\ 6 & 4 & -1 & 1 \end{array} \right] \) |
| \(\) | \(= \) | \( -2 \det \left[ \begin{array}{rrrr} 1 & 3 & 1 & 1 \\ 0 & 1 & 3 & 4 \\ 2 & -1 & -1 & 2 \\ 3 & 4 & -1 & 1 \end{array} \right] \) |
| \(\) | \(= \) | \( -2 \det \left[ \begin{array}{rrrr} 1 & 3 & 1 & 1 \\ 0 & 1 & 3 & 4 \\ 0 & -7 & -3 & 0 \\ 0 & -5 & -4 & -2 \end{array} \right] \) |
| \(\) | \(= \) | \( -2 \det \left[ \begin{array}{rrrr} 1 & 3 & 1 & 1 \\ 0 & 1 & 3 & 4 \\ 0 & 0 & 18 & 28 \\ 0 & 0 & 11 & 18 \end{array} \right] \) |
| \(\) | \(= \) | \( -2 \det \left[ \begin{array}{rr} 18 & 28 \\ 11 & 18 \end{array} \right] \) |
| \(\) | \(=\) | \(-2(324-308) = -32\) |
We know that \(\det(\Mtrx{A}) = \det(\Mtrx{A}^T)\), a process which turns a matrix in upper triangular form into one in lower triangular form and vice versa. So, we may compute the determinant of a matrix by transforming it into lower triangular form, and then forming the product of its diagonal entries. As an example: compute the determinant of the matrix
\[ \Mtrx{B} = \left[ \begin{array}{rrrrr} 0 & -2 & -8 & -6 & -12 \\ 1 & 0 & 0 & 1 & 2 \\ 3 & 0 & -3 & -4 & 6 \\ 1 & 2 & 1 & 0 & 2 \\ 2 & 1 & 4 & 3 & 10 \end{array} \right] \]| \(\det (\Mtrx{B})\) | \(=\) | \( \det \left[ \begin{array}{rrrrr} 0 & -2 & -8 & -6 & -12 \\ 1 & 0 & 0 & 1 & 2 \\ 3 & 0 & -3 & -4 & 6 \\ 1 & 2 & 1 & 0 & 2 \\ 2 & 1 & 4 & 3 & 10 \end{array} \right] \) |
| \(\) | \(= \) | \( -\det \left[ \begin{array}{rrrrr} 0 & -6 & -8 & -2 & -12 \\ 1 & 1 & 0 & 0 & 2 \\ 3 & -4 & -3 & 0 & 6 \\ 1 & 0 & 1 & 2 & 2 \\ 2 & 3 & 4 & 1 & 10 \end{array} \right] \) |
| \(\) | \(= \) | \( -\det \left[ \begin{array}{rrrrr} 0 & -6 & -8 & -2 & -12 \\ 1 & 1 & 0 & 0 & 0 \\ 3 & -4 & -3 & 0 & 0 \\ 1 & 0 & 1 & 2 & 0 \\ 2 & 3 & 4 & 1 & 6 \end{array} \right] \) |
| \(\) | \(= \) | \( -\det \left[ \begin{array}{rrrrr} 4 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 3 & -4 & -3 & 0 & 0 \\ 1 & 0 & 1 & 2 & 0 \\ 2 & 3 & 4 & 1 & 6 \end{array} \right] \) |
| \(\) | \(=\) | \(144\) |