Chapter 3

Suppose that \(A\) and \(B\) are \(4 \times 4\) invertible matrices. If \(\operatorname{det}(A)=-2\) and \(\operatorname{det}(B)=3,\) compute each determinant below. $$\operatorname{det}\left(B^{2} A^{-1}\right)$$.

Find \(\operatorname{det}(A) .\) If \(A\) is invertible, use the adjoint method to find \(A^{-1}\). $$A=\left[\begin{array}{rrr}3 & 4 & 7 \\\2 & 6 & 1 \\\3 & 14 & -1\end{array}\right].$$

Use the cofactor expansion theorem to evaluate the given determinant along the specified row or column. \(\left|\begin{array}{rrr}3 & 1 & 4 \\ 7 & 1 & 2 \\ 2 & 3 & -5\end{array}\right|,\) columm 1.

Determine whether the given expression is a term in the determinant of order \(5 .\) If it is, determine whether the permutation of the column indices has even or odd parity and hence find whether the term has a plus or a minus sign attached to it. $$a_{13} a_{25} a_{31} a_{44} a_{42}$$.

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form. $$\left|\begin{array}{rrrr} 2 & -1 & 3 & 4 \\ 7 & 1 & 2 & 3 \\ -2 & 4 & 8 & 6 \\ 6 & -6 & 18 & -24 \end{array}\right|$$

Suppose that \(A\) and \(B\) are \(4 \times 4\) invertible matrices. If \(\operatorname{det}(A)=-2\) and \(\operatorname{det}(B)=3,\) compute each determinant below. $$\operatorname{det}(A B)$$.

Find \(\operatorname{det}(A) .\) If \(A\) is invertible, use the adjoint method to find \(A^{-1}\). $$A=\left[\begin{array}{rrr}2 & 5 & 7 \\\4 & -3 & 2 \\\6 & 9 & 11\end{array}\right].$$

Use the cofactor expansion theorem to evaluate the given determinant along the specified row or column. \(\left|\begin{array}{rrr}0 & 2 & -3 \\ -2 & 0 & 5 \\ 3 & -5 & 0\end{array}\right|,\) row 3.

Determine the values of the indices \(p\) and \(q\) such that the following are terms in a determinant of order \(4 .\) In each case, determine the number of inversions in the permutation of the column indices and hence find the appropriate sign that should be attached to each term. $$a_{21} a_{3 q} a_{p 2} a_{43}$$.

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form. $$\left|\begin{array}{rrrr} 7 & -1 & 3 & 4 \\ 14 & 2 & 4 & 6 \\ 21 & 1 & 3 & 4 \\ -7 & 4 & 5 & 8 \end{array}\right|$$

Suppose that \(A\) and \(B\) are \(4 \times 4\) invertible matrices. If \(\operatorname{det}(A)=-2\) and \(\operatorname{det}(B)=3,\) compute each determinant below. $$\operatorname{det}\left((-A)^{3}\left(2 B^{2}\right)\right)$$.

Find \(\operatorname{det}(A) .\) If \(A\) is invertible, use the adjoint method to find \(A^{-1}\). $$A=\left[\begin{array}{rrrr}5 & -1 & 2 & 1 \\\3 & -1 & 4 & 5 \\\1 & -1 & 2 & 1 \\\5 & 9 & -3 & 2 \end{array}\right].$$

Use the cofactor expansion theorem to evaluate the given determinant along the specified row or column. \(\left|\begin{array}{rrrr}1 & -2 & 3 & 0 \\ 4 & 0 & 7 & -2 \\ 0 & 1 & 3 & 4 \\\ 1 & 5 & -2 & 0\end{array}\right|,\) column 4

Determine the values of the indices \(p\) and \(q\) such that the following are terms in a determinant of order \(4 .\) In each case, determine the number of inversions in the permutation of the column indices and hence find the appropriate sign that should be attached to each term. $$a_{13} a_{p 4} a_{32} a_{2 q}$$

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form. $$\left|\begin{array}{rrrrr} 3 & 7 & 1 & 2 & 3 \\ 1 & 1 & -1 & 0 & 1 \\ 4 & 8 & -1 & 6 & 6 \\ 3 & 7 & 0 & 9 & 4 \\ 8 & 16 & -1 & 8 & 12 \end{array}\right|$$

Suppose that \(A\) and \(B\) are \(4 \times 4\) invertible matrices. If \(\operatorname{det}(A)=-2\) and \(\operatorname{det}(B)=3,\) compute each determinant below. $$\operatorname{det}\left(\left(\left(A^{-1} B\right)^{T}\right)\left(2 B^{-1}\right)\right)$$.

Find \(\operatorname{det}(A) .\) If \(A\) is invertible, use the adjoint method to find \(A^{-1}\). $$A=\left[\begin{array}{rrrr}-1 & 0 & 0 & 4 \\\1 & 1 & 0 & -2 \\\6 & 0 & -1 & -2 \\ -3 & 1 & 3 & 2\end{array}\right].$$

Use the cofactor expansion theorem to evaluate the given determinant along the specified row or column. \(\left|\begin{array}{rrrl}-3 & 0 & -1 & 0 \\ 0 & 4 & 0 & 2 \\ 1 & 4 & -4 & 2 \\\ 0 & 2 & 5 & 0\end{array}\right|,\) column 1.

Determine the values of the indices \(p\) and \(q\) such that the following are terms in a determinant of order \(4 .\) In each case, determine the number of inversions in the permutation of the column indices and hence find the appropriate sign that should be attached to each term. $$a_{p q} a_{34} a_{13} a_{42}$$.

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form. $$\begin{array}{|lllll|} 1 & 2 & 3 & 4 & 5 \\ 5 & 4 & 3 & 2 & 1 \\ 2 & 3 & 4 & 5 & 6 \\ 6 & 5 & 4 & 3 & 2 \\ 0 & 2 & 4 & 6 & 8 \end{array} |$$

Use Cramer's rule to determine the unique solution for \(x\) to the system \(A x=b\) for the given matrix \(A\) and vector \(\mathbf{b}\). $$A=\left[\begin{array}{rl}2 & 8 \\\\-2 & 4\end{array}\right], \mathbf{b}=\left[\begin{array}{r}0 \\\\-3\end{array}\right].$$

Evaluate the given determinant by using the Cofactor Expansion Theorem. Do not apply elementary row operations. $$\left|\begin{array}{rrr} -4 & 2 & -1 \\ 7 & -3 & 2 \\ -6 & 6 & 2 \end{array}\right|$$

Use Theorem 3.2 .5 to determine whether the given matrix is invertible or not. $$\left[\begin{array}{rr} -1 & 1 \\ 1 & -1 \end{array}\right]$$

Let $$A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 2 & 1 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 1 \\ 5 & -2 \\ 4 & 7 \end{array}\right], \quad C=\left[\begin{array}{rrr} 1 & 0 & 5 \\ 3 & -1 & 4 \\ 2 & -2 & 6 \end{array}\right]$$. Compute the determinants, where possible. $$\operatorname{det}(B)$$

Use Cramer's rule to determine the unique solution for \(x\) to the system \(A x=b\) for the given matrix \(A\) and vector \(\mathbf{b}\). $$A=\left[\begin{array}{ll}3 & 5 \\\6 & 2\end{array}\right], \mathbf{b}=\left[\begin{array}{l}4 \\ 9\end{array}\right].$$

Evaluate the given determinant by using the Cofactor Expansion Theorem. Do not apply elementary row operations. $$\left|\begin{array}{ccc} 1 & 0 & -2 \\ 3 & 1 & -1 \\ 7 & 2 & 5 \end{array}\right|$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rr}0 & -2 \\ 5 & 1\end{array}\right]\).

Use Theorem 3.2 .5 to determine whether the given matrix is invertible or not. $$\left[\begin{array}{ll} 2 & 1 \\ 3 & 2 \end{array}\right]$$

Let $$A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 2 & 1 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 1 \\ 5 & -2 \\ 4 & 7 \end{array}\right], \quad C=\left[\begin{array}{rrr} 1 & 0 & 5 \\ 3 & -1 & 4 \\ 2 & -2 & 6 \end{array}\right]$$. Compute the determinants, where possible. $$\operatorname{det}(C)$$

Use Cramer's rule to determine the unique solution for \(x\) to the system \(A x=b\) for the given matrix \(A\) and vector \(\mathbf{b}\). $$A=\left[\begin{array}{lr}\cos t & \sin t \\\\\sin t & -\cos t\end{array}\right], \mathbf{b}=\left[\begin{array}{c}e^{-t} \\\3 e^{-t}\end{array}\right].$$

Evaluate the given determinant by using the Cofactor Expansion Theorem. Do not apply elementary row operations. $$\left|\begin{array}{rrr} -1 & 2 & 3 \\ 0 & 1 & 4 \\ 2 & -1 & 3 \end{array}\right|$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rr}6 & -3 \\ -5 & -1\end{array}\right]\).

Use Theorem 3.2 .5 to determine whether the given matrix is invertible or not. $$\left[\begin{array}{rrr} -1 & 2 & 3 \\ 5 & -2 & 1 \\ 8 & -2 & 5 \end{array}\right]$$

Let $$A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 2 & 1 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 1 \\ 5 & -2 \\ 4 & 7 \end{array}\right], \quad C=\left[\begin{array}{rrr} 1 & 0 & 5 \\ 3 & -1 & 4 \\ 2 & -2 & 6 \end{array}\right]$$. Compute the determinants, where possible. $$\operatorname{det}\left(C^{T}\right)$$

Use Cramer's rule to determine the unique solution for \(x\) to the system \(A x=b\) for the given matrix \(A\) and vector \(\mathbf{b}\). $$A=\left[\begin{array}{rrr}4 & 1 & 3 \\\2 & -1 & 5 \\\2 & 3 & 1\end{array}\right], \mathbf{b}=\left[\begin{array}{l}5 \\\7 \\\2\end{array}\right].$$

Evaluate the given determinant by using the Cofactor Expansion Theorem. Do not apply elementary row operations. $$\left|\begin{array}{rrr} 2 & -1 & 3 \\ 5 & 2 & 1 \\ 3 & -3 & 7 \end{array}\right|$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rr}-4 & 7 \\ 1 & 7\end{array}\right]\).

Use Theorem 3.2 .5 to determine whether the given matrix is invertible or not. $$\left[\begin{array}{rrr} 2 & 6 & -1 \\ 3 & 5 & 1 \\ 2 & 0 & 1 \end{array}\right]$$

Use Cramer's rule to determine the unique solution for \(x\) to the system \(A x=b\) for the given matrix \(A\) and vector \(\mathbf{b}\). $$A=\left[\begin{array}{rrr}5 & 3 & 6 \\\2 & 4 & -7 \\\2 & 5 & 9\end{array}\right], \mathbf{b}=\left[\begin{array}{r}3 \\\\-1 \\\4\end{array}\right].$$

Evaluate the given determinant by using the Cofactor Expansion Theorem. Do not apply elementary row operations. $$\left|\begin{array}{rrr} 0 & -2 & 1 \\ 2 & 0 & -3 \\ -1 & 3 & 0 \end{array}\right|$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{lr}2 & -3 \\ 1 & 5\end{array}\right]\).

Use Theorem 3.2 .5 to determine whether the given matrix is invertible or not. $$\left[\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ -1 & 1 & -1 & 1 \\ 1 & 1 & -1 & -1 \\ -1 & 1 & 1 & -1 \end{array}\right]$$

Let $$A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 2 & 1 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 1 \\ 5 & -2 \\ 4 & 7 \end{array}\right], \quad C=\left[\begin{array}{rrr} 1 & 0 & 5 \\ 3 & -1 & 4 \\ 2 & -2 & 6 \end{array}\right]$$. Compute the determinants, where possible. $$\operatorname{det}(B A)$$

Use Cramer's rule to determine the unique solution for \(x\) to the system \(A x=b\) for the given matrix \(A\) and vector \(\mathbf{b}\). $$A=\left[\begin{array}{lll}3.1 & 3.5 & 7.1 \\\2.2 & 5.2 & 6.3 \\\1.4 & 8.1 & 0.9 \end{array}\right], \mathbf{b}=\left[\begin{array}{l}3.6 \\\2.5 \\\9.3\end{array}\right].$$

Evaluate the given determinant by using the Cofactor Expansion Theorem. Do not apply elementary row operations. $$\left|\begin{array}{rrrr} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 0 & -1 & 0 \\ 0 & 1 & 0 & 1 \end{array}\right|$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rr}9 & -8 \\ -7 & -3\end{array}\right]\).

Use Theorem 3.2 .5 to determine whether the given matrix is invertible or not. $$\left[\begin{array}{rrrr} 1 & 0 & 2 & -1 \\ 3 & -2 & 1 & 4 \\ 2 & 1 & 6 & 2 \\ 1 & -3 & 4 & 0 \end{array}\right]$$

Let $$A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 2 & 1 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 1 \\ 5 & -2 \\ 4 & 7 \end{array}\right], \quad C=\left[\begin{array}{rrr} 1 & 0 & 5 \\ 3 & -1 & 4 \\ 2 & -2 & 6 \end{array}\right]$$. Compute the determinants, where possible. $$\operatorname{det}\left(B^{T} A^{T}\right)$$

If \(A\) is an invertible \(n \times n\) matrix, prove property \(\mathrm{P} 9:\) $$\operatorname{det}\left(A^{-1}\right)=\frac{1}{\operatorname{det}(A)}.$$

Evaluate the given determinant by using the Cofactor Expansion Theorem. Do not apply elementary row operations. $$\left|\begin{array}{rrrl} 2 & -1 & 3 & 1 \\ 1 & 4 & -2 & 3 \\ 0 & 2 & -1 & 0 \\ 1 & 3 & -2 & 4 \end{array}\right|$$