Determinant Properties: Homework

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Section 4.6 Determinant Properties: Homework

Exercises Exercises

1.

Prove Theorem 4.4.4.

2.

Prove Corollary 4.4.7.

3.

Prove Theorem 4.4.9.

4.

Prove Theorem 4.4.10.

5.

Use the method of elimination as described in Remark 4.4.16 to find the determinant of the following matrices.

\(\displaystyle A=\lmatrix{rr} 10 \amp 3\\ -2 \amp \frac{1}{2} \rmatrix.\)
\(\displaystyle B=\lmatrix{rrr} 0 \amp 5 \amp 1 \\3 \amp 2 \amp 8\\ 6 \amp -1 \amp 2 \rmatrix.\)
\(\displaystyle C=\lmatrix{rrr} 6 \amp 0 \amp 3 \\2 \amp 1 \amp -1\\ 1 \amp 2 \amp 5 \rmatrix.\)
\(\displaystyle D=\lmatrix{rrr} 3 \amp 7 \amp 2 \\-6 \amp 1 \amp 4\\ 6 \amp 29 \amp 12 \rmatrix.\)
\(\displaystyle E=\lmatrix{rrr} 0 \amp 0 \amp -1 \\1 \amp 0 \amp 0\\ 0 \amp -1 \amp 0 \rmatrix.\)
\(F=\lmatrix{rrrr} 2 \amp 1 \amp 3 \amp 17 \\4 \amp 2 \amp 6 \amp -8\\ 5 \amp 1 \amp -3 \amp 2 \\ 1 \amp -1 \amp 1 \amp 0 \rmatrix.\)
Hint.
It might be helpful to start by swapping rows 1 and 4.
\(\displaystyle G=\lmatrix{rrrrr} 0 \amp 1 \amp 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp 0 \amp 0 \\ 1 \amp 0 \amp 0 \amp 0 \amp 0 \\0 \amp 0 \amp 0 \amp 0 \amp 1 \\0 \amp 0 \amp 0 \amp 1 \amp 0 \rmatrix.\)
\(\displaystyle H=\lmatrix{rrrr} 1 \amp -1 \amp 2 \amp -3 \\6 \amp 7 \amp 1 \amp 5\\ 15 \amp 11 \amp 8 \amp 1 \\ 7 \amp 6 \amp 9 \amp 3 \rmatrix.\)

6.

Prove or disprove: If \(A,B\in\R^{n\times n}\) satisfy \(\det(AB)\ne0\text{,}\)then both \(A\) and \(B\) are invertible.

7.

Let \(A=\begin{pmatrix}39\amp -180\\8\amp -37 \end{pmatrix} \text{.}\) Use Theorem 4.4.15 to find the two numbers \(\lambda_1,\lambda_2\in\R\) for which \(A-\lambda_i I\) is singular, and for each of the \(\lambda_i\) verify that \(A-\lambda_i I\) is singular.