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Section 4.2 Determinant Fundamentals: In-Class Practice
Worksheet 4.2.1 Worksheet
Use the formula in
Theorem 4.1.2 to find the determinant of the following matrices.
\(\displaystyle A=\lmatrix{rr} -1 \amp 4\\ 2 \amp -3\rmatrix\)
\(\displaystyle B=\lmatrix{rr} 4 \amp -2\\ 6 \amp -1\rmatrix\)
\(\displaystyle C=\lmatrix{rr} 1 \amp 2\\ 3 \amp 6\rmatrix\)
Multiply to find the matrix \(D=AB\text{.}\) Then find \(\det(D)\text{.}\) Do you notice a relationship to \(\det(A)\) and \(\det(B)\text{?}\)
Find the determinant for each matrix \(A\text{.}\)
\(\displaystyle A=\lmatrix{rrr} 7 \amp 0 \amp -3 \\ -5 \amp 1 \amp 8\\ 2 \amp -2 \amp 1 \rmatrix\)
\(\displaystyle A=\lmatrix{rrr}5 \amp 0 \amp 2\\-1 \amp 3 \amp -1\\2 \amp 4 \amp 0\rmatrix\)
\(\displaystyle A=\lmatrix{rrr}3 \amp 1 \amp 2\\0 \amp -1 \amp 2\\0 \amp 0 \amp 1\rmatrix\)
\(\displaystyle A=\lmatrix{rrr}1 \amp 2 \amp 3\\0 \amp 3 \amp -1\\2 \amp 4 \amp 6\rmatrix\)
Let
\(P\) be a permutation matrix (see
Definition 3.7.11 . Show that
\(\det(P)=\pm 1\)
Suppose \(A \in \R^{n \times n}\) is a matrix so that \(\det(A)=2\text{.}\) Let \(B=3A.\) Find \(\det(B)\) when
\(\displaystyle n=2\)
\(\displaystyle n=3\)
\(\displaystyle n=k\)
Without doing any computations show that the determinant of matrix
\(B\) in
Problem 2.d is zero. (Hint: Also use
Theorem 4.4.2 .)
Let \(A=\lmatrix{rrr} 3 \amp 2 \amp 5\\9 \amp 4 \amp 10\\ 6 \amp 2 \amp 12 \rmatrix.\)
Find \(\det(A).\)
Find \(\det(A^{T})\)
What relationship do you notice about \(\det(A)\) and \(\det(A^{T})\text{?}\)
Use elimination to reduce \(A\) to a matrix \(U\) in row-echelon form. Then find \(\det(U)\text{.}\)
What relationship do you notice about \(\det(A)\) and \(\det(U)\text{?}\)
Use formulas from this section to find the determinant of \(A\)
\(\displaystyle A=\lmatrix{rrrr} 1 \amp 2 \amp -1 \amp 0 \\3 \amp 5 \amp 2 \amp 3\\
-5 \amp -4 \amp 1 \amp 0\\
7 \amp -2 \amp 5 \amp -6 \rmatrix\)
\(A=\lmatrix{rrrr} -5 \amp 0 \amp 0 \amp 0\\ 0 \amp 4 \amp 0 \amp 0 \\ 0 \amp 0 \amp -2 \amp 0 \\ 0 \amp 0 \amp 0 \amp -1 \rmatrix\text{.}\)
Find a shortcut for computing the determinant in the last part and use it to find the determinant of \(A=\lmatrix{rrrrr}8 \amp 0 \amp 0 \amp 0 \amp 0\\ 0 \amp -1 \amp 0 \amp 0\amp 0\\ 0 \amp 0 \amp 5 \amp 0 \amp 0\\0 \amp 0 \amp 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 0 \amp 0 \amp 2 \rmatrix\)
