Chapter 3

Let \(A=\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]\) and assume \(\operatorname{det}(A)=-6 .\) Find \(\operatorname{det}(B)\) $$B=\left[\begin{array}{rrr} g & h & i \\ -2 d & -2 e & -2 f \\ -a & -b & -c \end{array}\right]$$

True or False: Given any real number \(r\) and any \(3 \times 3\) matrix \(A\) whose entries are all nonzero, it is always possible to change at most one entry of \(A\) to get a matrix \(B\) with \(\operatorname{det}(B)=r.\)

Determine the eigenvalues of the given matrix \(A\). That is, determine the scalars \(\lambda\) such that \(\operatorname{det}(A-\lambda I)=0.\) $$A=\left[\begin{array}{rrr} 2 & 0 & 0 \\ -1 & -6 & 0 \\ 3 & 3 & 7 \end{array}\right]$$

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

$$\text { Let } A=\left[\begin{array}{lll} 1 & 2 & 4 \\ 3 & 1 & 6 \\ k & 3 & 2 \end{array}\right]$$. (a) Find all value(s) of \(k\) for which the matrix \(A\) fails to be invertible. (b) In terms of \(k,\) determine the volume of the parallelepiped determined by the row vectors of the matrix A. Is that the same as the volume of the parallelepiped determined by the column vectors of the matrix \(A\) ? Explain how you know this without any calculation.

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

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrrr}0 & 0 & 0 & -3 \\ 0 & 0 & -7 & -1 \\ 0 & 2 & 6 & 9 \\ 1 & 8 & -8 & -9\end{array}\right]\).

Let \(A=\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]\) and assume \(\operatorname{det}(A)=-6 .\) Find \(\operatorname{det}(B)\) $$B=\left[\begin{array}{ccc} d & e & f \\ -3 a & -3 b & -3 c \\ g-4 d & h-4 e & i-4 f \end{array}\right]$$

Determine the eigenvalues of the given matrix \(A\). That is, determine the scalars \(\lambda\) such that \(\operatorname{det}(A-\lambda I)=0.\) $$A=\left[\begin{array}{rrr} 6 & 0 & -2 \\ 0 & 7 & 0 \\ -5 & 0 & -3 \end{array}\right]$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrrr}4 & 1 & 8 & 6 \\ 0 & -2 & 13 & 5 \\ 0 & 0 & -6 & 1 \\ 0 & 0 & 0 & -3\end{array}\right]\).

Let \(A=\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]\) and assume \(\operatorname{det}(A)=-6 .\) Find \(\operatorname{det}(B)\) $$B=\left[\begin{array}{ccc} 2 a & 2 d & 2 g \\ b-c & e-f & h-i \\ c-a & f-d & i-g \end{array}\right]$$

Determine the eigenvalues of the given matrix \(A\). That is, determine the scalars \(\lambda\) such that \(\operatorname{det}(A-\lambda I)=0.\) $$A=\left[\begin{array}{rrr} -5 & -5 & 0 \\ -8 & 1 & 0 \\ -5 & 3 & 7 \end{array}\right]$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrrr}-2 & 0 & 1 & 6 \\ -1 & 3 & -1 & -4 \\ 2 & 1 & 0 & 3 \\ 0 & 5 & -4 & -2\end{array}\right]\).

Let \(A\) and \(B\) be \(4 \times 4\) matrices such that \(\operatorname{det}(A)=5\) and \(\operatorname{det}(B)=3 .\) Compute the determinant of the given matrix. $$A B^{T}$$

Let \(A\) and \(B\) be \(n \times n\) matrices such that \(A B=-B A\) Use determinants to prove that if \(n\) is odd, then \(A\) and \(B\) cannot both be invertible.

Determine the eigenvalues of the given matrix \(A\). That is, determine the scalars \(\lambda\) such that \(\operatorname{det}(A-\lambda I)=0.\) $$A=\left[\begin{array}{rrr} -5 & -1 & 1 \\ -2 & -1 & 0 \\ -5 & 2 & 3 \end{array}\right]$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrrr}-2 & -1 & 4 & -6 \\ 0 & 1 & 0 & 2 \\ 0 & -6 & 3 & 2 \\ 0 & 8 & 5 & 1\end{array}\right]\).

Let \(A\) and \(B\) be \(4 \times 4\) matrices such that \(\operatorname{det}(A)=5\) and \(\operatorname{det}(B)=3 .\) Compute the determinant of the given matrix. $$A^{2} B^{5}$$

A real \(n \times n\) matrix \(A\) is called orthogonal if \(A A^{T}=\) \(A^{T} A=I_{n} .\) If \(A\) is an orthogonal matrix prove that \(\operatorname{det}(A)=\pm 1.\)

Find (a) \(\operatorname{det}(A),\) (b) the matrix of cofactors \(M_{C},(\mathrm{c})\) adj \((A),\) and, if possible, \((\mathrm{d}) A^{-1}.\) $$A=\left[\begin{array}{ll} 3 & 1 \\ 4 & 5 \end{array}\right]$$

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

Let \(A\) and \(B\) be \(4 \times 4\) matrices such that \(\operatorname{det}(A)=5\) and \(\operatorname{det}(B)=3 .\) Compute the determinant of the given matrix. $$\left(A^{-1} B^{2}\right)^{3}$$

Use Cramer's rule to solve the given linear system. $$\begin{array}{r} -3 x_{1}+x_{2}=3, \\ x_{1}+2 x_{2}=1. \end{array}$$

Find (a) \(\operatorname{det}(A),\) (b) the matrix of cofactors \(M_{C},(\mathrm{c})\) adj \((A),\) and, if possible, \((\mathrm{d}) A^{-1}.\) $$A=\left[\begin{array}{rr} -1 & -2 \\ 4 & 1 \end{array}\right]$$

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

Let \(A\) and \(B\) be \(4 \times 4\) matrices such that \(\operatorname{det}(A)=5\) and \(\operatorname{det}(B)=3 .\) Compute the determinant of the given matrix. $$(2 B)^{-1}(A B)^{T}$$

Use Cramer's rule to solve the given linear system. $$\begin{aligned} &2 x_{1}-x_{2}+x_{3}=2,\\\ &\begin{array}{l} 4 x_{1}+5 x_{2}+3 x_{3}=0, \\ 4 x_{1}-3 x_{2}+3 x_{3}=2. \end{array} \end{aligned}$$

Find (a) \(\operatorname{det}(A),\) (b) the matrix of cofactors \(M_{C},(\mathrm{c})\) adj \((A),\) and, if possible, \((\mathrm{d}) A^{-1}.\) $$A=\left[\begin{array}{rr} 5 & 2 \\ -15 & -6 \end{array}\right]$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{lllll}1 & 2 & 0 & 0 & 0 \\ 3 & 4 & 0 & 0 & 0 \\ 0 & 0 & 5 & 0 & 0 \\ 0 & 0 & 0 & 6 & 7 \\ 0 & 0 & 0 & 8 & 9\end{array}\right]\).

Let \(A\) and \(B\) be \(4 \times 4\) matrices such that \(\operatorname{det}(A)=5\) and \(\operatorname{det}(B)=3 .\) Compute the determinant of the given matrix. $$(5 A)(2 B)$$

Use Cramer's rule to solve the given linear system. $$\begin{aligned} 3 x_{1}+x_{2}+2 x_{3} &=-1, \\ 2 x_{1}-x_{2}+x_{3} &=-1, \\ 5 x_{2}+5 x_{3} &=-5. \end{aligned}$$

Find (a) \(\operatorname{det}(A),\) (b) the matrix of cofactors \(M_{C},(\mathrm{c})\) adj \((A),\) and, if possible, \((\mathrm{d}) A^{-1}.\) $$A=\left[\begin{array}{rrr} 2 & -3 & 0 \\ 2 & 1 & 5 \\ 0 & -1 & 2 \end{array}\right]$$

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

Let \(A\) and \(B\) be \(4 \times 4\) matrices such that \(\operatorname{det}(A)=5\) and \(\operatorname{det}(B)=3 .\) Compute the determinant of the given matrix. $$B^{-1} A^{-1}$$

Find (a) \(\operatorname{det}(A),\) (b) the matrix of cofactors \(M_{C},(\mathrm{c})\) adj \((A),\) and, if possible, \((\mathrm{d}) A^{-1}.\) $$A=\left[\begin{array}{rrr} -2 & 3 & -1 \\ 2 & 1 & 5 \\ 0 & 2 & 3 \end{array}\right]$$

Evaluate the determinant of the given matrix function. \(A(t)=\left[\begin{array}{cc}e^{6 t} & e^{4 t} \\ 6 e^{6 t} & 4 e^{4 t}\end{array}\right]\).

Let \(A\) and \(B\) be \(4 \times 4\) matrices such that \(\operatorname{det}(A)=5\) and \(\operatorname{det}(B)=3 .\) Compute the determinant of the given matrix. $$B^{-1}(2 A) B^{T}$$

Find (a) \(\operatorname{det}(A),\) (b) the matrix of cofactors \(M_{C},(\mathrm{c})\) adj \((A),\) and, if possible, \((\mathrm{d}) A^{-1}.\) $$A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 3 & -1 & 4 \\ 5 & 1 & 7 \end{array}\right]$$

Evaluate the determinant of the given matrix function. \(A(t)=\left[\begin{array}{lll}\sin t & \cos t & 1 \\ \cos t & -\sin t & 0 \\\ \sin t & -\cos t & 0\end{array}\right]\).

Let \(A\) and \(B\) be \(4 \times 4\) matrices such that \(\operatorname{det}(A)=5\) and \(\operatorname{det}(B)=3 .\) Compute the determinant of the given matrix. $$(4 B)^{3}$$

Find (a) \(\operatorname{det}(A),\) (b) the matrix of cofactors \(M_{C} (\mathrm{c})\) adj \((A),\) and, if possible, \((\mathrm{d}) A^{-1}.\) $$A=\left[\begin{array}{rrr} 0 & 1 & 2 \\ -1 & -1 & 3 \\ 1 & -2 & 1 \end{array}\right]$$

Evaluate the determinant of the given matrix function. \(A(t)=\left[\begin{array}{ccc}e^{2 t} & e^{3 t} & e^{-4 t} \\ 2 e^{2 t} & 3 e^{3 t} & -4 e^{-4 t} \\ 4 e^{2 t} & 9 e^{3 t} & 16 e^{-4 t}\end{array}\right]\).

Let \(A, B,\) and \(S\) be \(n \times n\) matrices. If \(S^{-1} A S=B\) must \(A=B ?\) Must \(\operatorname{det}(A)=\operatorname{det}(B) ?\) Justify your answers.

Evaluate the determinant of the given matrix function. \(A(t)=\left[\begin{array}{rrr}e^{-t} & e^{-5 t} & e^{2 t} \\ -e^{-t} & -5 e^{-5 t} & 2 e^{2 t} \\ e^{-t} & 25 e^{-5 t} & 4 e^{2 t}\end{array}\right]\).

Let $$A=\left[\begin{array}{lll} 1 & 2 & 4 \\ 3 & 1 & 6 \\ k & 3 & 2 \end{array}\right]$$ (a) In terms of \(k,\) find the volume of the parallelepiped determined by the row vectors of the matrix \(A\) (b) Does your answer to (a) change if we instead consider the volume of the parallelepiped determined by the column vectors of the matrix \(A ?\) Why or why not? (c) For what value(s) of \(k,\) if any, is \(A\) invertible?

Find (a) \(\operatorname{det}(A),\) (b) the matrix of cofactors \(M_{C},(\mathrm{c})\) adj \((A),\) and, if possible, \((\mathrm{d}) A^{-1}.\) $$A=\left[\begin{array}{rrr} 2 & -3 & 5 \\ 1 & 2 & 1 \\ 0 & 7 & -1 \end{array}\right]$$

Find (a) \(\operatorname{det}(A),\) (b) the matrix of cofactors \(M_{C},(\mathrm{c})\) adj \((A),\) and, if possible, \((\mathrm{d}) A^{-1}.\) $$A=\left[\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ -1 & 1 & -1 & 1 \\ 1 & 1 & -1 & -1 \\ -1 & 1 & 1 & -1 \end{array}\right]$$

We explore a relationship between determinants and solutions to a differential equation. The \(3 \times 3\) matrix consisting of solutions to a differential equation and their derivatives is called the Wronskian and, as we will see in later chapters, plays a pivotal role in the theory of differential equations. Verify that \(y_{1}(x)=\cos 2 x, y_{2}(x)=\sin 2 x,\) and \(y_{3}(x)=e^{x}\) are solutions to the differential equation \(y^{\prime \prime \prime}-y^{\prime \prime}+4 y^{\prime}-4 y=0\) and show that \(\left|\begin{array}{lll}y_{1} & y_{2} & y_{3} \\\ y_{1}^{\prime} & y_{2}^{\prime} & y_{3}^{\prime} \\ y_{1}^{\prime \prime} & y_{2}^{\prime \prime} & y_{3}^{\prime \prime}\end{array}\right|\) is nonzero on any interval.

Without expanding the determinant, determine all values of \(x\) for which \(\operatorname{det}(A)=0\) if $$A=\left[\begin{array}{rrr} 1 & -1 & x \\ 2 & 1 & x^{2} \\ 4 & -1 & x^{3} \end{array}\right]$$

Find (a) \(\operatorname{det}(A),\) (b) the matrix of cofactors \(M_{C},(\mathrm{c})\) adj \((A),\) and, if possible, \((\mathrm{d}) A^{-1}.\) $$A=\left[\begin{array}{rrrr} 1 & 0 & 3 & 5 \\ -2 & 1 & 1 & 3 \\ 3 & 9 & 0 & 2 \\ 2 & 0 & 3 & -1 \end{array}\right]$$