Objectives: Topics
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The relationships between row reductions, invertibility, and determinants.
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Section 3.2 Properties of the Determinant
Handout 3.2 Properties of the Determinant
Objectives: Goals
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Apply determinant properties (related to row reductions, transpose, and matrix products) to compute determinants efficiently.
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Use determinants to determine whether a square matrix is invertible.
We have a method for computing determinants, but without the strategies in this section, the cofactor-expansion algorithm can be very inefficient. Row operations provide a more efficient method.
Theorem 3.10. Row Operations and the Determinant.
Let \(A\) be a square matrix.
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If a multiple of one row of \(A\) is added to another to produce \(B\text{,}\) then \(\operatorname{det}(B)=\operatorname{det}(A)\text{.}\)
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If two rows of \(A\) are interchanged to produce \(B\text{,}\) then \(\operatorname{det}(B)=-\operatorname{det}(A)\text{.}\)
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If a row of \(A\) is multiplied by a scalar \(k\) to produce \(B\text{,}\) then \(\operatorname{det}(B)=k\operatorname{det}(A)\text{.}\)
Example 3.11.
Compute:
\begin{equation*}
\begin{vmatrix}
1 \amp -4 \amp 2\\
-2 \amp 8 \amp -9\\
-1 \amp 7 \amp 3
\end{vmatrix}
\end{equation*}
Example 3.12.
Compute:
\begin{equation*}
\begin{vmatrix}
2 \amp 6 \amp -5 \amp 2\\
2 \amp 5 \amp -7 \amp 3\\
4 \amp 15 \amp -4 \amp 6\\
-2 \amp -5 \amp 4 \amp 2
\end{vmatrix}
\end{equation*}
Theorem 3.13.
\begin{equation*}
|A|=
\begin{cases}
(-1)^r \times \text{(product of pivots)}, \amp \text{if $A$ is invertible},\\[4pt]
0, \amp \text{if $A$ is singular}.
\end{cases}
\end{equation*}
Theorem 3.14.
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\(A\) is invertible if and only if \(\operatorname{det}(A)\neq 0\text{.}\)
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\(\operatorname{det}(AB)=\operatorname{det}(A)\operatorname{det}(B)\text{.}\)
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\(\operatorname{det}(A)=\operatorname{det}(A^T)\text{.}\)
Example 3.15.
Determine the value of:
\begin{equation*}
\operatorname{det}(A)
=
\operatorname{det}
\left(
\begin{bmatrix}
0 \amp 2 \amp 0\\
1 \amp 1 \amp 2\\
1 \amp 1 \amp 3
\end{bmatrix}^8
\right).
\end{equation*}
Example 3.16.
Use the determinant to find all values of \(\lambda\) for which the matrix \(C\) is not invertible.
\begin{equation*}
C=
\begin{bmatrix}
5 \amp 0 \amp 0\\
0 \amp 0 \amp 1\\
1 \amp 1 \amp 0
\end{bmatrix}
-
\lambda I_3.
\end{equation*}
