91Ó°ÊÓ

Show that the scalar and matrices \(\alpha=-3, \quad A=\left(\begin{array}{rr}-2 & 4 \\ 4 & 0\end{array}\right), \quad\) and \(\quad B=\left(\begin{array}{rr}0 & -5 \\ -2 & 3\end{array}\right)\) satisfy the given identity. \(\left(A^{T}\right)^{T}=A\)

Which of the matrices are singular? If a matrix is nonsingular, find its inverse. \(A=\left(\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right)\)

Calculate the determinant of the given matrix. Determine if the matrix has a nontrivial nullspace, and if it does find a basis for the nullspace. Determine if the column vectors in the matrix are linearly independent. \(\left(\begin{array}{rr}-1 & 2 \\ 2 & -4\end{array}\right)\)

Sketch the nullspace of the given matrix in \(\mathbf{R}^{2}\). \(A=\left(\begin{array}{rr}1 & 2 \\ -2 & -4\end{array}\right)\)

If you have a computer or calculator that will place an augmented matrix in reduced row echelon form, use it to help find the solution of each system \(A \mathbf{y}=\mathbf{b}\) given. Otherwise you'll have to do the calculations by hand. \(A=\left(\begin{array}{ccc}-6 & 8 & 0 \\ 4 & 8 & 8 \\ -2 & 2 & 7\end{array}\right) \quad\) and \(\quad \mathbf{b}=\left(\begin{array}{c}2 \\\ 20 \\ 7\end{array}\right)\)

Which of the matrices are singular? If a matrix is nonsingular, find its inverse. \(A=\left(\begin{array}{lll}1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 2 & 1\end{array}\right)\)

Show that the scalar and matrices \(\alpha=-3, \quad A=\left(\begin{array}{rr}-2 & 4 \\ 4 & 0\end{array}\right), \quad\) and \(\quad B=\left(\begin{array}{rr}0 & -5 \\ -2 & 3\end{array}\right)\) satisfy the given identity. \((\alpha A)^{T}=\alpha A^{T}\)

Sketch the nullspace of the given matrix in \(\mathbf{R}^{2}\). \(A=\left(\begin{array}{ll}2 & 3 \\ 4 & 6\end{array}\right)\)

If you have a computer or calculator that will place an augmented matrix in reduced row echelon form, use it to help find the solution of each system \(A \mathbf{y}=\mathbf{b}\) given. Otherwise you'll have to do the calculations by hand. \(A=\left(\begin{array}{ccc}3 & -3 & 1 \\ 7 & -4 & 5 \\ 4 & -3 & -3\end{array}\right) \quad\) and \(\quad \mathbf{b}=\left(\begin{array}{l}0 \\\ 3 \\ 1\end{array}\right)\)

Calculate the determinant of the given matrix. Determine if the matrix has a nontrivial nullspace, and if it does find a basis for the nullspace. Determine if the column vectors in the matrix are linearly independent. \(\left(\begin{array}{rr}1 & 3 \\ -1 & -3\end{array}\right)\)