Write each matrix equation as a system of linear equations without matrices. [ 2 0 − 1 0 3 0 1 1 0 ] [ x y z ] = [ 6 9 5 ] \begin{bmatrix}2 & 0 & -1 \\0 & 3 & 0 \\1 & 1 & 0\end{bmatrix}\begin{bmatrix}x \\y \\z\end{bmatrix}=\begin{bmatrix}6 \\9 \\5\end{bmatrix} 2 0 1 0 3 1 − 1 0 0 x y z = 6 9 5

express the following matrix equation as a system of equations. And we have this matrix right here. Now we're going to rewrite this to this system of equations to do this. We're going to multiply these two matrices on the left and set it equal to the matrix on the right. So we notice here we have 4 -20 X, Y. Z. So with multiplying matrices we're gonna multiply the first part of our row, our top row by the first part of our column. In this case we have four top row. That's gonna multiply the X. And the same for the next part negative. Two will multiply the Y and zero. All multiply. This is set equal to 10. Now we can do the same for our middle row. You have zero times x A six times away. Plus seven. Time to see equals four for our last one. We have zero times X. Eight times one Plus zero. Time to see that's equal tonight. We can rewrite this entire system of equations As four X -2. Y equals 10. Six, Y plus seven. Z equals four and eight. Y equals nine. If you look at our possible answers, we can tell that the answer to our problem is going to be answer. B. Alright, I hope to help you solve the problem. Thank you for watching. Goodbye

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

College Algebra

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

1:59 minutes

Problem 35

Textbook Question

Write each matrix equation as a system of linear equations without matrices.
$\begin{bmatrix}\)2 & 0 & -1 \\0 & 3 & 0 \\1 & 1 & 0$\end{bmatrix}\[\begin{bmatrix}$x \$\y$ \$\z\]\end{bmatrix}$=$\begin{bmatrix}$6 \\9 \\5\(\end{bmatrix}$

Verified step by step guidance

Identify the matrix equation given: $\begin{bmatrix} 2 & 0 & -1 \\ 0 & 3 & 0 \\ 1 & 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 9 \\ 5 \end{bmatrix}$.

Recall that multiplying a matrix by a vector corresponds to taking the dot product of each row of the matrix with the vector. This results in a system of linear equations.

Write the first row multiplied by the vector: $2x + 0 \cdot y + (-1)z = 6$, which simplifies to \$2x - z = 6$.

Write the second row multiplied by the vector: $0 \cdot x + 3y + 0 \cdot z = 9$, which simplifies to \$3y = 9$.

Write the third row multiplied by the vector: $1x + 1y + 0 \cdot z = 5$, which simplifies to $x + y = 5$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Matrix Multiplication

Matrix multiplication involves multiplying each row of the first matrix by each column of the second matrix and summing the products. In this problem, multiplying the coefficient matrix by the variable matrix results in a new matrix representing the system's left-hand side expressions.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same variables. Writing the matrix equation as a system means expressing each row multiplication as an individual linear equation equated to the corresponding constant.

Variables and Constants in Matrix Form

The variables (x, y, z) are represented as a column matrix, and the constants on the right side form another column matrix. Understanding how these correspond to the coefficients and constants in the system is essential for translating between matrix and equation forms.

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Master Determinants of 3×3 Matrices with a bite sized video explanation from Patrick

Write each matrix equation as a system of linear equations without matrices.[20−1030110][xyz]=[695]\(\begin{bmatrix}\)2 & 0 & -1 \\0 & 3 & 0 \\1 & 1 & 0\(\end{bmatrix}\[\begin{bmatrix}\)x \\(\y\) \\(\z\]\end{bmatrix}\)=\(\begin{bmatrix}\)6 \\9 \\5\(\end{bmatrix}\)201031−100xyz=695

Key Concepts

Matrix Multiplication

System of Linear Equations

Variables and Constants in Matrix Form

Watch next

Write each matrix equation as a system of linear equations without matrices.
$\begin{bmatrix}\)2 & 0 & -1 \\0 & 3 & 0 \\1 & 1 & 0\(\end{bmatrix}\[\begin{bmatrix}\)x \\(\y\) \\(\z\]\end{bmatrix}\)=\(\begin{bmatrix}\)6 \\9 \\5\(\end{bmatrix}$