In each of the following, express the matrix equation as a...

In each of the following, express the matrix equation as a system of linear equations. (a) $\left[\begin{array}{ccc}3 & -1 & 2 \\ 4 & 3 & 7 \\ -2 & 1 & 5\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right]=\left[\begin{array}{c}2 \\ -1 \\ 4\end{array}\right]$ $$ \text { (b) }\left[\begin{array}{cccc} 3 & -2 & 0 & 1 \\ 5 & 0 & 2 & -2 \\ 3 & 1 & 4 & 7 \\ -2 & 5 & 1 & 6 \end{array}\right]\left[\begin{array}{l} w \\ x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] $$

In each of the following, express the matrix equation as a system of linear equations.
(a) $\left[\begin{array}{ccc}3 & -1 & 2 \\ 4 & 3 & 7 \\ -2 & 1 & 5\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right]=\left[\begin{array}{c}2 \\ -1 \\ 4\end{array}\right]$
$$
\text { (b) }\left[\begin{array}{cccc}
3 & -2 & 0 & 1 \\
5 & 0 & 2 & -2 \\
3 & 1 & 4 & 7 \\
-2 & 5 & 1 & 6
\end{array}\right]\left[\begin{array}{l}
w \\
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
0 \\
0 \\
0 \\
0
\end{array}\right]
$$

Key Concepts

Matrix Equation

A matrix equation expresses a system of linear equations in a compact form, where a coefficient matrix multiplies a variable vector to yield a constant or result vector. This approach simplifies the representation and manipulation of multiple equations that share the same coefficients and variables.

Matrix Multiplication

Matrix multiplication is the process used to combine the coefficient matrix with the vector of variables. In this operation, each entry in the resulting vector is the sum of the products of corresponding entries from a row of the matrix and the variable vector, effectively reflecting one linear equation per row.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. Each equation represents a linear combination of these variables equal to a constant. Converting a matrix equation into a system of equations involves writing out each row of the matrix multiplication as an individual equation.

Homogeneous and Non-Homogeneous Systems

Systems of linear equations can be classified as homogeneous or non-homogeneous. A homogeneous system has all constant terms equal to zero, while a non-homogeneous system has at least one non-zero constant term. This classification affects the characteristics of the solution set, including the possibility of non-trivial solutions in the homogeneous case.

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