91Ó°ÊÓ

Fill in the blank. A rectangular array of real numbers that can be used to solve a system of linear equations is called a _________________ .

Two matrices are _____ when they have the same dimension and all of their corresponding entries are equal.

Do all square matrices have inverses?

When a system of linear equations has no solution, do the lines intersect?

Is a consistent system with infinitely many solutions independent or dependent?

Equality of Matrices Find \(x\) and \(y\) or \(x, y,\) and \(z.\) $$\left[\begin{array}{rrr} x+4 & 8 & -3 \\ 1 & 22 & 2 y \\ 7 & -2 & z+2 \end{array}\right]=\left[\begin{array}{rrr} 2 x+9 & 8 & -3 \\ 1 & 22 & -8 \\ 7 & -2 & 11 \end{array}\right]$$

Operations with Matrices Find, if possible, \((a) A+B,(b) A-B,(c) 3 A,\) and \((d) 3 A-2 B.\) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{rr} 8 & -1 \\ 2 & 3 \\ -4 & 5 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 6 \\ -1 & -5 \\ 1 & 10 \end{array}\right]$$

Operations with Matrices Find, if possible, \((a) A+B,(b) A-B,(c) 3 A,\) and \((d) 3 A-2 B.\) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{rrr} -1 & 4 & 0 \\ 3 & -2 & 2 \\ 5 & 4 & -1 \\ 0 & 8 & -6 \\ -4 & -1 & 0 \end{array}\right], B=\left[\begin{array}{rrr} -3 & 5 & 1 \\ 2 & -4 & -7 \\ 10 & -9 & -1 \\ 3 & 2 & -4 \\ 0 & 1 & -2 \end{array}\right]$$

Identify the elementary row operation performed to obtain the new row- equivalent matrix. New Row-Equivalent Matrix \(\left[\begin{array}{rrr}1 & -2 & 5 \\ -2 & 6 & 7\end{array}\right]\) \(\left[\begin{array}{rrr}-18 & 0 & 6 \\ 5 & 2 & -2\end{array}\right]\) \(\left[\begin{array}{rrrr}-1 & -2 & 3 & -2 \\ 2 & -5 & 1 & -7 \\ 0 & -6 & 8 & -4\end{array}\right]\) \(\left[\begin{array}{rrrr}-1 & 3 & -7 & 6 \\ 0 & -1 & -5 & 5 \\ 4 & -5 & 1 & 3\end{array}\right]\) Original Matrix \(\left[\begin{array}{ccc}-4 & 8 & -20 \\ -2 & 6 & 7\end{array}\right]\)

Solve the system graphically. Verify your solutions algebraically. $$\left\\{\begin{aligned} -x-y &=3 \\ x^{2}+y^{2}-4 x-21 &=0 \end{aligned}\right.$$