91Ó°ÊÓ

Make the indicated changes in the determinant at the right, and then solve the indicated problem. Assume the elements are nonzero, unless otherwise specified. $$\left|\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right|$$ Evaluate the determinant if \(a=c, d=f,\) and \(g=i\)

Solve the indicated systems of equations using the inverse of the coefficient matrix. In Exercises \(35-40,\) it is necessary to set up the appropriate equations. For the following system of equations, solve for \(x^{2}\) and \(y\) using the matrix methods of this section, and then solve for \(x\) and \(y\) $$\begin{aligned} x^{2}+y &=2 \\ 2 x^{2}-y &=10 \end{aligned}$$

Find \(B A^{-1} .\) In Exercises \(32-34,\) find \(C A^{-1}\) $$B=\left[\begin{array}{ll} 8 & -2 \\ 3 & 4 \end{array}\right]$$ $$C=\left[\begin{array}{rrr} 5 & -1 & 0 \\ 2 & -2 & 1 \\ -3 & 0 & 4 \end{array}\right]$$ $$A=\left[\begin{array}{rr} 2 & -4 \\ -1 & 3 \end{array}\right]$$

Make the indicated changes in the determinant at the right, and then solve the indicated problem. Assume the elements are nonzero, unless otherwise specified. $$\left|\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right|$$ Evaluate the determinant if \(b=c=f=0\)

Solve the given problems using Gaussian elimination. Solve the system \(x+2 y=6,2 x+a y=4\) and show that the solution depends on the value of \(a\). What value of \(a\) does the solution show may not be used?

Find \(B A^{-1} .\) In Exercises \(32-34,\) find \(C A^{-1}\) $$B=\left[\begin{array}{ll} 8 & -2 \\ 3 & 4 \end{array}\right]$$ $$C=\left[\begin{array}{rrr} 5 & -1 & 0 \\ 2 & -2 & 1 \\ -3 & 0 & 4 \end{array}\right]$$ $$A=\left[\begin{array}{rr} -4 & 1 \\ 6 & -2 \end{array}\right]$$

Determine by matrix multiplication whether or not A is the proper matrix of solution values. $$\begin{array}{l} 2 x-y+z=7 \\ x-3 y+2 z=7 \\ 3 x+y=7 \end{array} \quad A=\left[\begin{array}{r} 3 \\ -2 \\ -1 \end{array}\right]$$

Solve the indicated systems of equations using the inverse of the coefficient matrix. In Exercises \(35-40,\) it is necessary to set up the appropriate equations. For the following system of equations, solve for \(x^{2}\) and \(y^{2}\) using the matrix methods of this scction, and then solve for \(x\) and \(y\) $$\begin{aligned} &x^{2}-y^{2}=8\\\ &x^{2}+y^{2}=10 \end{aligned}$$

Perform the indicated matrix multiplications on a calculator, using the following matrices. For matrix \(A, A^{2}=A \times A\) $$A=\left[\begin{array}{rrr} 2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 & -4 \end{array}\right] B=\left[\begin{array}{rrr} 1 & -2 & -6 \\ -3 & 2 & 9 \\ 2 & 0 & -3 \end{array}\right] C=\left[\begin{array}{rrr} 1 & -3 & -4 \\ -1 & 3 & 4 \\ 1 & -3 & -4 \end{array}\right]$$

Find \(B A^{-1} .\) In Exercises \(32-34,\) find \(C A^{-1}\) $$B=\left[\begin{array}{ll} 8 & -2 \\ 3 & 4 \end{array}\right]$$ $$C=\left[\begin{array}{rrr} 5 & -1 & 0 \\ 2 & -2 & 1 \\ -3 & 0 & 4 \end{array}\right]$$ $$A=\left[\begin{array}{ll} 5 & -3 \\ 2 & -1 \end{array}\right]$$