To solve this problem, let’s consider what this matrix might represent. This matrix appears to be an augmented matrix representing a system of linear equations. The matrix form is often used to represent multiple equations in a compact and manageable way.
A matrix [tex]\( \left[\begin{array}{cccc}a & b & c & d \\ e & f & g & h \\ i & j & k & l\end{array}\right] \)[/tex] of size [tex]\(3 \times 4\)[/tex] typically corresponds to a system of three linear equations with three unknowns. Here, the matrix form is:
[tex]\[
\left[\begin{array}{cccc}1 & -3 & -1 & 400 \\ -3 & 10 & 3 & 300 \\ 0 & 6 & 1 & 150\end{array}\right]
\][/tex]
This augmented matrix can be translated into the following system of linear equations:
1. [tex]\(1x - 3y - 1z = 400\)[/tex]
2. [tex]\(-3x + 10y + 3z = 300\)[/tex]
3. [tex]\(0x + 6y + 1z = 150\)[/tex]
Now, let's interpret and outline the meaning of each row:
### Row 1:
The equation from the first row is:
[tex]\[ x - 3y - 1z = 400 \][/tex]
### Row 2:
The equation from the second row is:
[tex]\[ -3x + 10y + 3z = 300 \][/tex]
### Row 3:
The equation from the third row is:
[tex]\[ 0x + 6y + 1z = 150 \][/tex]
which simplifies to:
[tex]\[ 6y + z = 150 \][/tex]
These equations form a system of linear equations to be solved for the variables [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex].
### Summary:
We have successfully translated the given augmented matrix into a system of linear equations:
[tex]\[
\begin{cases}
x - 3y - z = 400 \\
-3x + 10y + 3z = 300 \\
6y + z = 150
\end{cases}
\][/tex]
The solution process would typically involve using methods such as substitution, elimination, or matrix operations like Gaussian elimination to find the values of [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex].
