Chapter 3

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rr}2 & -4 \\ -1 & 0\end{array}\right]\).

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

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. $$\operatorname{det}(B A C)$$

If \(A\) is an arbitrary \(3 \times 3\) matrix, use cofactor expansion to show that property P5 holds: $$\operatorname{det}\left(A^{T}\right)=\operatorname{det}(A).$$

Evaluate the given determinant by using the Cofactor Expansion Theorem. Do not apply elementary row operations. $$\left|\begin{array}{rrrr} -4 & 1 & -3 & -2 \\ 1 & 2 & -1 & -6 \\ 3 & 0 & 2 & 3 \\ 0 & 0 & -5 & 2 \end{array}\right|$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rr}1 & 4 \\ -4 & 3\end{array}\right]\).

Determine all values of the constant \(k\) for which the given system has an infinite number of solutions. $$\begin{aligned}x_{1}+2 x_{2}+k x_{3} &=0 \\ 2 x_{1}-k x_{2}+x_{3} &=0 \\ 3 x_{1}+6 x_{2}+x_{3} &=0 \end{aligned}$$

Assume that \(A\) and \(B\) be \(3 \times 3\) matrices with \(\operatorname{det}(A)=3\) and \(\operatorname{det}(B)=-4 .\) Compute the specified determinant. \(\operatorname{det}(2 A)\)

Use elementary row operations together with the Cofactor Expansion Theorem to evaluate the given determinant. $$\left|\begin{array}{rrr} -1 & 3 & 3 \\ 4 & -6 & -3 \\ 2 & -1 & 4 \end{array}\right|$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{ll}e^{-3} & 3 e^{10} \\ 2 e^{-5} & 6 e^{8}\end{array}\right]\).

Determine all values of the constant \(k\) for which the given system has a unique solution $$\begin{aligned} x_{1}+k x_{2} &=b_{1} \\ k x_{1}+4 x_{2} &=b_{2} \end{aligned}$$

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. $$\operatorname{det}\left(\left(A A^{T}\right)^{2}\right)$$

Assume that \(A\) and \(B\) be \(3 \times 3\) matrices with \(\operatorname{det}(A)=3\) and \(\operatorname{det}(B)=-4 .\) Compute the specified determinant. \(\operatorname{det}\left(A^{-1}\right)\)

Use elementary row operations together with the Cofactor Expansion Theorem to evaluate the given determinant. $$\left|\begin{array}{rrrr} 3 & 5 & 2 & 6 \\ 2 & 3 & 5 & -5 \\ 7 & 5 & -3 & -16 \\ 9 & -6 & 27 & -12 \end{array}\right|$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{ll}\pi & \pi^{2} \\ \sqrt{2} & 2 \pi\end{array}\right]\).

Determine all values of \(k\) for which the given system has a unique solution. $$\begin{aligned} x_{1}+k x_{2} &=2 \\ k x_{1}+x_{2}+x_{3} &=1 \\ x_{1}+x_{2}+x_{3} &=1 \end{aligned}$$

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. $$\operatorname{det}\left(C^{-1} B A\right)$$

Use elementary row operations together with the Cofactor Expansion Theorem to evaluate the given determinant. $$\left|\begin{array}{rrrr} 2 & -7 & 4 & 3 \\ 5 & 5 & -3 & 7 \\ 6 & 2 & 6 & 3 \\ 4 & 2 & -4 & 5 \end{array}\right|$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrr}6 & -1 & 2 \\ -4 & 7 & 1 \\ 0 & 3 & 1\end{array}\right]\).

Determine all values of \(k\) for which the given system has an infinite number of solutions. $$\begin{aligned} x_{1}+2 x_{2}+x_{3} &=k x_{1} \\ 2 x_{1}+x_{2}+x_{3} &=k x_{2} \\ x_{1}+x_{2}+2 x_{3} &=k x_{3} \end{aligned}$$

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. $$\operatorname{det}\left(B C C^{T}\right)$$

Assume that \(A\) and \(B\) be \(3 \times 3\) matrices with \(\operatorname{det}(A)=3\) and \(\operatorname{det}(B)=-4 .\) Compute the specified determinant. \(\operatorname{det}\left(B^{5}\right)\)

Use elementary row operations together with the Cofactor Expansion Theorem to evaluate the given determinant. $$\left|\begin{array}{rrrr} -2 & 0 & 1 & 1 \\ 1 & 2 & 2 & 0 \\ -4 & 4 & 6 & 1 \\ -1 & 1 & 0 & 5 \end{array}\right|$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrr}5 & -3 & 0 \\ 1 & 4 & -1 \\ -8 & 2 & -2\end{array}\right]\).

If \(A=\left[\begin{array}{rrr}1 & -1 & 2 \\ 3 & 1 & 4 \\ 0 & 1 & 3\end{array}\right],\) find \(\operatorname{det}(A),\) and use properties of determinants to find \(\operatorname{det}\left(A^{-1}\right)\) and \(\operatorname{det}(-3 A)\)

Let \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right],\) and let \(B=\left[\begin{array}{ll}5 & 4 \\ 1 & 1\end{array}\right] .\) Use the adjoint method to find \(B^{-1}\) and then determine \(\left(A^{-1} B^{T}\right)^{-1}.\)

Assume that \(A\) and \(B\) be \(3 \times 3\) matrices with \(\operatorname{det}(A)=3\) and \(\operatorname{det}(B)=-4 .\) Compute the specified determinant. \(\operatorname{det}\left(B^{-1} A B\right)^{2}\)

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrr}-2 & -4 & 1 \\ 6 & 1 & 1 \\ -2 & -1 & 3\end{array}\right]\).

Use the adjoint method to determine \(A^{-1}\) for the given matrix \(A.\) $$A=\left[\begin{array}{rrr} 2 & -1 & 1 \\ 0 & 5 & -1 \\ 1 & 1 & 3 \end{array}\right]$$.

Assume that \(A\) and \(B\) be \(3 \times 3\) matrices with \(\operatorname{det}(A)=3\) and \(\operatorname{det}(B)=-4 .\) Compute the specified determinant. \(\operatorname{det}(C),\) where \(C\) is obtained from matrix \(B\) by interchanging the last two columns and multiplying the first column by 4

Use elementary row operations together with the Cofactor Expansion Theorem to evaluate the given determinant. $$\begin{array}{|llrrl|} 2 & 0 & -1 & 3 & 0 \\ 0 & 3 & 0 & 1 & 2 \\ 0 & 1 & 3 & 0 & 4 \\ 1 & 0 & 1 & -1 & 0 \\ 3 & 0 & 2 & 0 & 5 \end{array} |$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrr}0 & 0 & -3 \\ 0 & 4 & 3 \\ -2 & 1 & 5\end{array}\right]\).

Use the adjoint method to determine \(A^{-1}\) for the given matrix \(A.\) $$A=\left[\begin{array}{rrrr} 0 & -3 & 2 & 2 \\ 0 & 1 & 1 & 1 \\ 1 & 2 & 3 & -4 \\ 1 & 0 & 0 & 5 \end{array}\right]$$.

If \(A=\left[\begin{array}{rrrr}0 & x & y & z \\ -x & 0 & 1 & -1 \\ -y & -1 & 0 & 1 \\ -z & 1 & -1 & 0\end{array}\right],\) show that \(\operatorname{det}(A)=\) \((x+y+z)^{2}.\)

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrr}9 & 1 & -7 \\ 6 & 2 & 1 \\ -4 & 0 & -2\end{array}\right]\).

Let \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) and assume \(\operatorname{det}(A)=1 .\) Find \(\operatorname{det}(B)\) $$B=\left[\begin{array}{cc} -2 a & -2 c \\ 3 a+b & 3 c+d \end{array}\right]$$

Use the adjoint method to determine \(A^{-1}\) for the given matrix \(A.\) $$A=\left[\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 1 & 3 & -3 \\ -2 & -3 & -5 & 2 \\ 4 & -4 & 4 & 6 \end{array}\right]$$.

(a) Consider the \(3 \times 3\) Vandermonde determinant $$ \begin{aligned} V\left(r_{1}, r_{2}, r_{3}\right) & \text { defined by } \\ V\left(r_{1}, r_{2}, r_{3}\right) &=\left|\begin{array}{ccc} 1 & 1 & 1 \\ r_{1} & r_{2} & r_{3} \\ r_{1}^{2} & r_{2}^{2} & r_{3}^{2} \end{array}\right| \end{aligned} $$ Show that $$ V\left(r_{1}, r_{2}, r_{3}\right)=\left(r_{2}-r_{1}\right)\left(r_{3}-r_{1}\right)\left(r_{3}-r_{2}\right) $$ (b) More generally, show that the \(n \times n\) Vandermonde determinant $$ V\left(r_{1}, r_{2}, \ldots, r_{n}\right)=\left|\begin{array}{cccc} 1 & 1 & \ldots & 1 \\ r_{1} & r_{2} & \ldots & r_{n} \\ r_{1}^{2} & r_{2}^{2} & \ldots & r_{n}^{2} \\ \vdots & \vdots & & \vdots \\ r_{1}^{n-1} & r_{2}^{n-1} & \ldots & r_{n}^{n-1} \end{array}\right| $$ has value $$ V\left(r_{1}, r_{2}, \ldots, r_{n}\right)=\prod_{1 \leq i

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrr}2 & -10 & 3 \\ 1 & 1 & 1 \\ 0 & 8 & -3\end{array}\right]\).

Let \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) and assume \(\operatorname{det}(A)=1 .\) Find \(\operatorname{det}(B)\) $$B=\left[\begin{array}{ll} 3 c & 3 d \\ 4 a & 4 b \end{array}\right]$$

Use the adjoint method to determine \(A^{-1}\) for the given matrix \(A.\) $$A=\left[\begin{array}{rrr} 5 & 8 & 16 \\ 4 & 1 & 8 \\ -4 & -4 & -11 \end{array}\right]$$.

Determine the eigenvalues of the given matrix \(A\). That is, determine the scalars \(\lambda\) such that \(\operatorname{det}(A-\lambda I)=0.\) $$A=\left[\begin{array}{rr} 2 & -1 \\ 2 & 4 \end{array}\right]$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrr}5 & 4 & 3 \\ -2 & 9 & 12 \\ 1 & -1 & 0\end{array}\right]\).

Let \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) and assume \(\operatorname{det}(A)=1 .\) Find \(\operatorname{det}(B)\) $$B=\left[\begin{array}{rr} -6 d & -6 c \\ 3 b & 3 a \end{array}\right]$$

Use the adjoint method to determine \(A^{-1}\) for the given matrix \(A.\) $$A=\left[\begin{array}{lll} 2 & 6 & 6 \\ 2 & 7 & 6 \\ 2 & 7 & 7 \end{array}\right]$$.

Determine the eigenvalues of the given matrix \(A\). That is, determine the scalars \(\lambda\) such that \(\operatorname{det}(A-\lambda I)=0.\) $$A=\left[\begin{array}{ll} 2 & 4 \\ 3 & 13 \end{array}\right]$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{lll}5 & 0 & 4 \\ 0 & 3 & 0 \\ 2 & 0 & 1\end{array}\right]\).

Add one row to the matrix \(A=\left[\begin{array}{rrr}4 & -1 & 0 \\ 5 & 1 & 4\end{array}\right]\) so as to create a \(3 \times 3\) matrix \(B\) with \(\operatorname{det}(B)=10.\)

Determine the eigenvalues of the given matrix \(A\). That is, determine the scalars \(\lambda\) such that \(\operatorname{det}(A-\lambda I)=0.\) $$A=\left[\begin{array}{ll} -1 & 2 \\ -4 & 7 \end{array}\right]$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{ccc}\sqrt{\pi} & e^{2} & e^{-1} \\ \sqrt{67} & 1 / 30 & 2001 \\ \pi & \pi^{2} & \pi^{3}\end{array}\right]\).