Chapter 3

Evaluate the determinant of the given matrix \(A\) by using (a) Definition \(3.1 .8,\) (b) elementary row operations to reduce \(A\) to an upper triangular matrix, and (c) the Cofactor Expansion Theorem. $$A=\left[\begin{array}{rr} 6 & 6 \\ -2 & 1 \end{array}\right]$$.

Evaluate the given determinant. $$|-3|$$

Determine all minors and cofactors of the given matrix. $$A=\left[\begin{array}{rr} -9 & 2 \\ 0 & 5 \end{array}\right]$$

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form. $$\left|\begin{array}{rr} -2 & 5 \\ 5 & -4 \end{array}\right|$$

Determine the number of inversions and the parity of the given permutation. (3,1,4,2).

Evaluate the determinant of the given matrix \(A\) by using (a) Definition \(3.1 .8,\) (b) elementary row operations to reduce \(A\) to an upper triangular matrix, and (c) the Cofactor Expansion Theorem. $$A=\left[\begin{array}{rr} -7 & -2 \\ 1 & -5 \end{array}\right]$$.

Evaluate the given determinant. $$\left|\begin{array}{rr}5 & -1 \\\3 & 7\end{array}\right|.$$

Determine all minors and cofactors of the given matrix. $$A=\left[\begin{array}{rr} 1 & -3 \\ 2 & 4 \end{array}\right]$$

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

Determine the number of inversions and the parity of the given permutation. (2,4,3,1).

Evaluate the determinant of the given matrix \(A\) by using (a) Definition \(3.1 .8,\) (b) elementary row operations to reduce \(A\) to an upper triangular matrix, and (c) the Cofactor Expansion Theorem. $$A=\left[\begin{array}{rrr} 2 & 3 & -5 \\ -4 & 0 & 2 \\ 6 & -3 & 3 \end{array}\right]$$.

Evaluate the given determinant. $$\left|\begin{array}{rrr}3 & 5 & 7 \\\\-1 & 2 & 4 \\\6 & 3 & -2\end{array}\right|.$$

Determine all minors and cofactors of the given matrix. $$A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 3 & -1 & 4 \\ 2 & 1 & 5 \end{array}\right]$$

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

Determine the number of inversions and the parity of the given permutation. (5,4,3,2,1).

Evaluate the determinant of the given matrix \(A\) by using (a) Definition \(3.1 .8,\) (b) elementary row operations to reduce \(A\) to an upper triangular matrix, and (c) the Cofactor Expansion Theorem. $$A=\left[\begin{array}{rrr} -1 & 4 & 1 \\ 0 & 2 & 2 \\ 2 & 2 & -3 \end{array}\right]$$.

Evaluate the given determinant. $$\left|\begin{array}{rrr}5 & 1 & 4 \\\6 & 1 & 3 \\\14 & 2 & 7\end{array}\right|.$$

Determine all minors and cofactors of the given matrix. $$A=\left[\begin{array}{rrr} 2 & 10 & 3 \\ 0 & -1 & 0 \\ 4 & 1 & 5 \end{array}\right]$$

Determine the number of inversions and the parity of the given permutation. (2,4,1,5,3).

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

Evaluate the determinant of the given matrix \(A\) by using (a) Definition \(3.1 .8,\) (b) elementary row operations to reduce \(A\) to an upper triangular matrix, and (c) the Cofactor Expansion Theorem. $$A=\left[\begin{array}{rrrr} 0 & 0 & 0 & -2 \\ 0 & 0 & -5 & 1 \\ 0 & 1 & -4 & 1 \\ -3 & -3 & -3 & -3 \end{array}\right]$$.

Evaluate the given determinant. $$\left|\begin{array}{ccc}2.3 & 1.5 & 7.9 \\\4.2 & 3.3 & 5.1 \\\6.8 & 3.6 & 5.7\end{array}\right|.$$

Determine all minors and cofactors of the given matrix. $$\text { If } A=\left[\begin{array}{rrrr} 1 & 3 & -1 & 2 \\ 3 & 4 & 1 & 2 \\ 7 & 1 & 4 & 6 \\ 5 & 0 & 1 & 2 \end{array}\right]$$ determine the minors \(\vec{M}_{12}, M_{31}, M_{23}, M_{42},\) and the corresponding cofactors.

Determine the number of inversions and the parity of the given permutation. (6,1,4,2,5,3).

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form. $$\left|\begin{array}{rrr} 0 & 1 & -2 \\ -1 & 0 & 3 \\ 2 & -3 & 0 \end{array}\right|$$

Evaluate the determinant of the given matrix \(A\) by using (a) Definition \(3.1 .8,\) (b) elementary row operations to reduce \(A\) to an upper triangular matrix, and (c) the Cofactor Expansion Theorem. $$A=\left[\begin{array}{rrrr} 3 & -1 & -2 & 1 \\ 0 & 0 & 1 & 4 \\ 0 & 2 & 1 & -1 \\ 0 & 0 & 0 & -4 \end{array}\right]$$.

Evaluate the given determinant. $$\left|\begin{array}{lll}a & b & c \\\b & c & a \\\c & a & b\end{array}\right|.$$

If $$ A=\left[\begin{array}{rrrr} -2 & 9 & 0 & -1 \\ 4 & -6 & 8 & 8 \\ 0 & -1 & -3 & 4 \\ 7 & -7 & 3 & 1 \end{array}\right] $$ determine the minors \(M_{41}, M_{22}, M_{23}, M_{43},\) and the corresponding cofactors.

Determine the number of inversions and the parity of the given permutation. (6,5,4,3,2,1).

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form. $$\left|\begin{array}{rrr} 3 & 7 & 1 \\ 5 & 9 & -6 \\ 2 & 1 & 3 \end{array}\right|$$

Suppose that $$A=\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right] \text { and } \operatorname{det}(A)=4$$. Compute the determinant of each matrix below. $$\left[\begin{array}{ccc} a-5 d & b-5 e & c-5 f \\ 3 g & 3 h & 3 i \\ -d+3 g & -e+3 h & -f+3 i \end{array}\right]$$.

Evaluate the given determinant. $$\left|\begin{array}{rrrr}3 & 5 & -1 & 2 \\\2 & 1 & 5 & 2 \\\3 & 2 & 5 & 7 \\\1 & -1 & 2 & 1 \end{array}\right|.$$

Use the cofactor expansion theorem to evaluate the given determinant along the specified row or column. \(\left|\begin{array}{cc}-8 & 6 \\ -4 & 9\end{array}\right|,\) column 2

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

Use Definition 3.1 .8 to derive the general expression for the determinant of \(A\) if $$A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right]$$.

Suppose that $$A=\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right] \text { and } \operatorname{det}(A)=4$$. Compute the determinant of each matrix below. $$\left[\begin{array}{rrr} g & h & i \\ -4 a & -4 b & -4 c \\ 2 d & 2 e & 2 f \end{array}\right]$$

Evaluate the given determinant. $$\left|\begin{array}{rrrr}7 & 1 & 2 & 3 \\\2 & -2 & 4 & 6 \\\3 & -1 & 5 & 4 \\\18 & 9 & 27 & 54 \end{array}\right|.$$

Use the cofactor expansion theorem to evaluate the given determinant along the specified row or column. \(\left|\begin{array}{rr}1 & -2 \\ 1 & 3\end{array}\right|,\) row 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. $$a_{11} a_{23} a_{34} a_{43} a_{52}$$.

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form. $$\left|\begin{array}{rrrr} 2 & 32 & 1 & 4 \\ 26 & 104 & 26 & -13 \\ 2 & 56 & 2 & 7 \\ 1 & 40 & 1 & 5 \end{array}\right|$$

Suppose that $$A=\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right] \text { and } \operatorname{det}(A)=4$$. Compute the determinant of each matrix below. $$3 \cdot\left[\begin{array}{ccc} a-d & b-e & c-f \\ 2 g & 2 h & 2 i \\ -d & -e & -f \end{array}\right]$$.

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

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

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. $$a_{11} a_{25} a_{33} a_{42} a_{54}$$.

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form. $$\left|\begin{array}{rrrr} 0 & 1 & -1 & 1 \\ -1 & 0 & 1 & 1 \\ 1 & -1 & 0 & 1 \\ -1 & -1 & -1 & 0 \end{array}\right|$$

Suppose that $$A=\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right] \text { and } \operatorname{det}(A)=4$$. Compute the determinant of each matrix below. $$\left[\begin{array}{ccc} 3 b & 3 e & 3 h \\ c-2 a & f-2 d & i-2 g \\ -a & -d & -g \end{array}\right]$$.

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

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

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. $$a_{11} a_{32} a_{24} a_{43} a_{55}$$.

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