91Ó°ÊÓ

Indicate whether the matrix is in rowreduced form. $$ \left[\begin{array}{lll|l} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 5 \end{array}\right] $$

Perform the indicated operations. $$ \left[\begin{array}{rrrr} 2 & -3 & 4 & -1 \\ 3 & 1 & 0 & 0 \end{array}\right]+\left[\begin{array}{rrrr} 4 & 3 & -2 & -4 \\ 6 & 2 & 0 & -3 \end{array}\right] $$

Compute the indicated products. $$ \left[\begin{array}{rr} -1 & 2 \\ 4 & 3 \\ 0 & 1 \end{array}\right]\left[\begin{array}{lll} 2 & 1 & 2 \\ 3 & 2 & 4 \end{array}\right] $$

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(x+2 y=3\) \(2 x-3 y=-8\) \(x-4 y=-9\)

Find the inverse of the matrix, if it exists. Verify your answer. $$ \left[\begin{array}{rrr} 3 & -2 & 7 \\ -2 & 1 & 4 \\ 6 & -5 & 8 \end{array}\right] $$

Determine the value of \(k\) for which the system of linear equations $$ \begin{array}{r} 3 x+4 y=12 \\ x+k y=4 \end{array} $$ has infinitely many solutions. Then find all solutions corresponding to this value of \(k\).

Indicate whether the matrix is in rowreduced form. $$ \left[\begin{array}{lll|l} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 2 & -3 \end{array}\right] $$

Matrix \(A\) is an input-output matrix associated with an economy, and matrix \(D\) (units in millions of dollars) is a demand vector. In each problem,find the final outputs of each industry such that the demands of industry and the consumer sector are met. $$ A=\left[\begin{array}{ll} 0.1 & 0.4 \\ 0.3 & 0.2 \end{array}\right] \text { and } D=\left[\begin{array}{r} 5 \\ 10 \end{array}\right] $$

Find the inverse of the matrix, if it exists. Verify your answer. $$ \left[\begin{array}{rrrr} 1 & 1 & -1 & 1 \\ 2 & 1 & 1 & 0 \\ 2 & 1 & 0 & 1 \\ 2 & -1 & -1 & 3 \end{array}\right] $$

Matrix \(A\) is an input-output matrix associated with an economy, and matrix \(D\) (units in millions of dollars) is a demand vector. In each problem,find the final outputs of each industry such that the demands of industry and the consumer sector are met. $$ A=\left[\begin{array}{lll} \frac{1}{5} & \frac{2}{5} & \frac{1}{5} \\ \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & \frac{1}{5} & 0 \end{array}\right] \text { and } D=\left[\begin{array}{r} 10 \\ 5 \\ 15 \end{array}\right] $$