91Ó°ÊÓ

Find the determinants of the matrices \(A\) and find out which of these matrices are invertible. $$\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{array}\right]$$

Use the determinant to find out for which values of the constant \(k\) the given matrix \(A\) is invertible. $$\left[\begin{array}{ll} k & 2 \\ 3 & 4 \end{array}\right]$$

Use the determinant to find out for which values of the constant \(k\) the given matrix \(A\) is invertible. $$\left[\begin{array}{ll} 1 & k \\ k & 4 \end{array}\right]$$

Consider a \(4 \times 4\) matrix \(A\) with rows \(\vec{v}_{1}, \vec{v}_{2}, \vec{v}_{3}, \vec{v}_{4}\) If \(\operatorname{det}(A)=8,\) find the determinants in Exercises 11 through 16. $$\operatorname{det}\left[\begin{array}{l} \vec{v}_{4} \\ \vec{v}_{2} \\ \vec{v}_{3} \\ \vec{v}_{1} \end{array}\right]$$

Use the determinant to find out for which values of the constant \(k\) the given matrix \(A\) is invertible. $$\left[\begin{array}{lll} k & 3 & 5 \\ 0 & 2 & 6 \\ 0 & 0 & 4 \end{array}\right]$$

Consider a \(4 \times 4\) matrix \(A\) with rows \(\vec{v}_{1}, \vec{v}_{2}, \vec{v}_{3}, \vec{v}_{4}\) If \(\operatorname{det}(A)=8,\) find the determinants in Exercises 11 through 16. $$\operatorname{det}\left[\begin{array}{c} \vec{v}_{1} \\ \vec{v}_{2}+9 \vec{v}_{4} \\ \vec{v}_{3} \\ \vec{v}_{4} \end{array}\right]$$

Use the determinant to find out for which values of the constant \(k\) the given matrix \(A\) is invertible. $$\left[\begin{array}{lll} 4 & 0 & 0 \\ 3 & k & 0 \\ 2 & 1 & 0 \end{array}\right]$$

Use the determinant to find out for which values of the constant \(k\) the given matrix \(A\) is invertible. $$\left[\begin{array}{lll} 0 & k & 1 \\ 2 & 3 & 4 \\ 5 & 6 & 7 \end{array}\right]$$

Use the determinant to find out for which values of the constant \(k\) the given matrix \(A\) is invertible. $$\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & k & 5 \\ 6 & 7 & 8 \end{array}\right]$$

True or false? If \(\Omega\) is a parallelogram in \(\mathbb{R}^{3}\) and \(T(\vec{x})=\) \(A \vec{x}\) is a linear transformation from \(\mathbb{R}^{3}\) to \(\mathbb{R}^{3}\), then area of \(T(\Omega)=|\operatorname{det} A|(\operatorname{area} \text { of } \Omega)\)