91Ó°ÊÓ

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

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 \(\mathbf{b}\). $$A=\left[\begin{array}{lr}\cos t & \sin t \\\\\sin t & -\cos t\end{array}\right], \mathbf{b}=\left[\begin{array}{c}e^{-t} \\\3 e^{-t}\end{array}\right].$$

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

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

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. \(A=\left[\begin{array}{lr}2 & -3 \\ 1 & 5\end{array}\right]\).

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