Use matrix multiplication to determine whether each matrix...

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

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

Key Concepts

Verification of Solutions

Verifying a solution to a system of equations involves substituting the proposed solution into the original equations to check if the equalities hold true. In the context of matrix multiplication, if the product of the coefficient matrix and a candidate vector equals the constant vector (b), then the candidate is a solution of the system.

Matrix Multiplication

Matrix multiplication is a method of combining a matrix and a vector (or another matrix) to produce a new vector (or matrix). This operation involves taking the dot product of rows of the matrix with the corresponding entries of the vector, which is central when using matrices to represent systems of equations. It provides a systematic way to evaluate whether a given vector satisfies the corresponding linear system when multiplied by the system’s coefficient matrix.

System of Linear Equations

A system of linear equations consists of multiple linear equations that are solved simultaneously. Each equation in the system represents a line (in two dimensions, for example), and the solution to the system is the point(s) at which these lines intersect. The matrix form of a system, Ax = b, conveys the same information, allowing for the use of linear algebra techniques to find solutions.

Matrix Representation of Systems

Systems of linear equations can be expressed in matrix form where the coefficient matrix (A) multiplies the variable vector (x) to produce a constant vector (b). This compact representation is useful not only for theoretical purposes but also for computational efficiency, particularly when using methods like matrix multiplication or algorithms designed for numerical solutions.

Graphing Utility for Verification

A graphing utility is a tool that can visually represent the linear equations of the system. By graphing the lines represented by each equation, one can visually confirm the point of intersection, which should correspond to the solution obtained through matrix multiplication. This serves as an effective means to validate the accuracy of computed solutions.

Transcript

00:01 So in this problem, we're given this system of equations, minus 2x minus 3y equals minus 6, and 4x plus 2y equals 20, and as to use matrix multiplication to determine whether each matrix is a solution of the system.

00:27 Okay, so if i think of this left, side in a matrix form, then for a, i can set up this matrix minus 2, minus 3, 4, 2, where i take the coefficients, times a, they give me the matrix 3 ,0, and do this multiplication.

00:57 So this is going to give me a 2x2 by 1 matrix, right? and so i'm going to have, minus 2 times 3, that's minus 6, minus 6 plus, or minus 3 times 0 is 0.

01:17 So that's minus 6.

01:20 4 times 3 is 12, and 2 times 0 is 0.

01:27 And so this is not equal this matrix of minus 620, right? because to be equal, every entry has to be the same, and those are not.

01:46 So that one didn't work.

01:47 All right.

01:48 So let's try the next one now.

01:50 Minus two, minus three, four, two.

01:55 And they give us matrix six negative two.

02:03 Okay.

02:06 So this becomes matrix six times negative two.

02:14 That's negative 12 plus negative three times negative two.

02:19 That's a positive six.