### Matrix calculus: Determinants

### More properties of determinants

Determinant of transpose and product

If #A# and #B# are both #( n\times n)#-matrices, then

\[\begin{array}{lrcl}\text{determinant of the transpose: }& \det (A)&=&\det (A^\top)\\\text{determinant of the product: } &\det (A\ B)&=&\det (A)\cdot\det (B)\end{array} \]

A special consequence of the above product formula is a criterion for invertibility of the matrix:

Invertibility in terms of determinant A square matrix #A# is invertible if and only if the determinant of #A# is distinct from #0#.

In this case, #\det(A^{-1}) = \frac{1}{\det(A)}#.

We discuss a few special cases.

Determinants of some special matrices

- The determinant of a square matrix of the form \[M = \matrix{A&C\\ 0&B}\] where #A# and #B# are square submatrices, and #C# is an arbitrary matrix of appropriate dimensions, is equal to the product of the determinants of the two submatrices along the diagonal: \[\det(M) = \det(A)\cdot\det(B)\]
- The determinant of a square matrix of the form \[M = \matrix{a_{11}&\cdots&\cdots&\cdots&a_{1n}\\ 0&a_{22}&\cdots&\cdots&a_{2n}\\ 0&0&\ddots&\vdots&a_{3n}\\ 0&0&\ddots&a_{(n-1)(n-1)}&a_{nn}\\ 0&0&\cdots&0&a_{nn}}\] is equal to the product of the diagonal entries: \[\det(M) = a_{11}\cdot a_{22}\,\cdots\, a_{nn}\]

*Later* we will see how these laws help to compute the determinant of a matrix efficiently.

#a=# #-3#

The variable #a# occurs in each entry of the first row, so

\[A = \matrix{a&0&0\\ 0&1&0\\ 0&0&1}\, \matrix{2 & -3 & 4 \\ 4 & -5 & 2 \\ 8 & 1 & 3 \\ }\] The determinant of the first matrix on the right-hand side is #a# and the determinant of the second is #130#. Consequently, the

The variable #a# occurs in each entry of the first row, so

\[A = \matrix{a&0&0\\ 0&1&0\\ 0&0&1}\, \matrix{2 & -3 & 4 \\ 4 & -5 & 2 \\ 8 & 1 & 3 \\ }\] The determinant of the first matrix on the right-hand side is #a# and the determinant of the second is #130#. Consequently, the

*product formula for the determinant*gives \[-390= a \cdot 130\] from which it immediately follows that #a = -3#.
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