Determinant

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Determinant

The determinant of a matrix is a map from the space of matrices to the real numbers: $$ \det GL(n) \rightarrow \mathbb{R} $$ The determinant is uniquely defined by the following two properties. $$ \det(I_n)= 1 $$ $$ \det(M)\times\det(N) = \det(MN) $$ In practice, we calculate the determinant recursively as follows

$$ \det(M) = m_{11}M_{1,1} - m_{12}M_{1,2} +(-1)^{n+1}\dots m_{1n}M_{1,n}$$

where $m_{ij}$ is the element of the matrix in row $i$ and column $j$ and $M_{i,j}$ denotes the determinant of the sub matrix of $M$ formed by deleting the $i^{th}$ row and $j^{th}$ column. This is also called the $(i,j)$ - minor.

This is an operation that is easier understood through example than through description!

 

Calculation of determinant 1

Find the determinant of the following matrix: $$M = \left(\begin{array}{ccc}1 &2 &4 \\ 3 & 5 & -2 \\ 2 & 5 & 2\end{array}\right)$$
solution - press button to display

The determinant of $M$ is given by

$$\begin{align} \det(M) &= 1\det\left(\begin{array}{cc}5 & -2 \\ 5 & 2\end{array}\right) - 2\det\left(\begin{array}{cc}3 & -2 \\ 2 & 2\end{array}\right) + 4\det\left(\begin{array}{cc}3 & 5 \\ 2 & 5\end{array}\right) \\ &= 1((5)(2) - (-2)(5)) - 2((3)(2) - (-2)(2)) + 4((3)(5) - (5)(2))\\ &= 20 - 20 +20 \\ &= 20\end{align}$$

Calculation of determinant 2

Calculate the determinant of the matrix $$ A = \left( \begin{array}{ccc} 2 & 4 & 5 \\ 0 & 0 & 3 \\ 4 & 1 & 5 \\ \end{array} \right) $$
solution - press button to display

The interesting aspect of this question is expanding the derivative about the second row instead of the first; this will substantially reduce the amount of work:

The determinant is given by $$ \begin{align} \det(A) &= -m_{21}M_{21} + m_{22}M_{22} - m_{23}M_{23} \\ &= 0\cdot M_{21} + 0\cdot M_{22} - 3M_{23} \\ &= -3((2)(1) - (4)(4)) \\ &= 42 \end{align} $$