In Exercises 49–52, use matrix multiplication to determine...

In Exercises 49–52, use matrix multiplication to determine whether each matrix is a solution of the system of equations. Use a graphing utility to verify your results. $$ \left\{\begin{aligned}-2 x-3 y &=-6 \\ 4 x+2 y &=20 \end{aligned}\right. $$ $$ \begin{array}{l}{\text { (a) }\left[\begin{array}{c}{3} \\ {0}\end{array}\right]} & {\text { (b) }\left[\begin{array}{r}{6} \\ {-2}\end{array}\right]} \\ {\begin{array}{ll}{\text { (c) }\left[\begin{array}{r}{-6} \\ {6}\end{array}\right]} & {\text { (d) }\left[\begin{array}{l}{4} \\ {2}\end{array}\right]}\end{array}}\end{array} $$

In Exercises 49–52, use matrix multiplication to determine whether each matrix is a solution of the system of equations. Use a graphing utility to verify your results.
$$
\left\{\begin{aligned}-2 x-3 y &=-6 \\ 4 x+2 y &=20 \end{aligned}\right.
$$
$$
\begin{array}{l}{\text { (a) }\left[\begin{array}{c}{3} \\ {0}\end{array}\right]} & {\text { (b) }\left[\begin{array}{r}{6} \\ {-2}\end{array}\right]} \\ {\begin{array}{ll}{\text { (c) }\left[\begin{array}{r}{-6} \\ {6}\end{array}\right]} & {\text { (d) }\left[\begin{array}{l}{4} \\ {2}\end{array}\right]}\end{array}}\end{array}
$$

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Key Concepts

Matrix Multiplication

Matrix multiplication is a core operation in linear algebra that combines rows of one matrix with columns of another to produce a new matrix or vector. It is especially important for evaluating the left-hand side of a system of linear equations when expressed in matrix form, enabling efficient computation and verification of solutions.

System of Linear Equations

A system of linear equations consists of multiple linear equations sharing common variables. Solving such systems involves finding variable values that satisfy all equations simultaneously. Representing these systems using matrices simplifies processes such as solution verification, computation, and analysis of the systems’ properties.

Solution Verification

Solution verification involves substituting candidate solutions into the equations or performing matrix multiplication with the candidate solution vector to check if the results match the expected outcomes on the right-hand side. This process confirms whether the candidate vector is, indeed, a valid solution to the system.

Graphical Verification

Graphical verification uses tools like graphing utilities to plot each equation in a system. The intersection point(s) of these plots visually confirm the solution(s) to the system, providing an alternative method to algebraic verification.

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