MATH1012 - Lecture 9

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MATH1012 - Lecture 8
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Determinant

• $\det (A^T)=\det (A)$
• If $A$ has a row of zeroes then $\det (A)=0$
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Computing the determinant of size $n\times n$ requires computing the determinant of $n$ matrices of order $n-1$

Upper Triangular

• An upper triangular matrix (all numbers below the diagonal are zero) has an easy determinant to compute, equal to the product of all values across diagonal
  • It is easy to solve determinants of matrices in row echelon form

Solving Determinant Using Row-Echelon Form

• When reducing a matrix to solve its determinant, keep track of the changes you make to the original matrix ($A$)
• Thus, when you solve the $\det (A^{\prime })$, you can transform it into $\det (A)$
• The effect of each change is here
  • $\det (A^{\prime })$ does not necessarily equal $\det (A)$!

Properties of Determinants

• $\det (\alpha A)={\alpha }^n\det (A)$
• $A$ is invertible if and only if $\det (A)\ne 0$

Determinant is Multiplicative

• $\det (AB)=\det (A)\times \det (B)=\det (AB)$
• $\det (A^k)=(\det (A){)}^k$
• $\det (A^{-1})=\frac{1}{\det (A)}$

Invertible Matrix Theorem

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