91Ó°ÊÓ

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. \(\left(\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0\end{array}\right)\)

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. \(A=\left(\begin{array}{ccc}-4 & 10 & -6 \\ 0 & -4 & 4 \\ 2 & -10 & 8\end{array}\right) \quad\) and \(\quad \mathbf{b}=\left(\begin{array}{c}-14 \\\ 4 \\ 12\end{array}\right)\)

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. \(\left(\begin{array}{rrr}1 & -2 & -4 \\ 2 & 1 & 2 \\ 3 & 0 & 0\end{array}\right)\)

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. \(A=\left(\begin{array}{ccc}-3 & -3 & 1 \\ 8 & 7 & -2 \\ 8 & 6 & -1\end{array}\right) \quad\) and \(\quad \mathbf{b}=\left(\begin{array}{c}4 \\\ -8 \\ -5\end{array}\right)\)

Describe the nullspace of the given matrix, both parametrically and geometrically in \(\mathbf{R}^{3}\). \(\left(\begin{array}{rrr}2 & 1 & -1 \\ 1 & -1 & -2 \\ 2 & 0 & -2\end{array}\right)\)

Consider \(S=\left\\{\left(\begin{array}{l}t \\ 0\end{array}\right) \mid t>0\right\\}\). Is \(S\) the solution set of a system of linear equations?

Which of the matrices are singular? If a matrix is nonsingular, find its inverse. \(A=\left(\begin{array}{rrrr}3 & -1 & -3 & -1 \\ 0 & -3 & -4 & 1 \\ -2 & 1 & 2 & 1 \\ 1 & 1 & 3 & -3\end{array}\right)\)

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. \(A=\left(\begin{array}{ccc}5 & 9 & 2 \\ -2 & -3 & -1 \\ 0 & -2 & 1\end{array}\right) \quad\) and \(\quad \mathbf{b}=\left(\begin{array}{c}8 \\\ -2 \\ -4\end{array}\right)\)

Without actually solving, which systems have unique solutions? Explain. \(x_{1}+2 x_{2}=4\) \(x_{1}-x_{2}=6\)

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. \(\left(\begin{array}{rrr}-1 & 0 & -1 \\ 1 & 1 & 2 \\ 2 & 1 & 3\end{array}\right)\)