Solving Matrix Equations: A Journey through Linear Algebra
Matrix equations are a powerful tool in linear algebra, allowing us to solve systems of linear
equations in a compact and elegant way. In this chapter, we explore various methods to
solve matrix equations, with examples drawn from the video tutorials.
Row Operations and Gaussian Elimination
One of the most fundamental methods for solving matrix equations is through row operations
and Gaussian elimination. This process involves transforming the given matrix into reduced
row echelon form (RREF), from which we can easily read off the solution.
For instance, consider the following system of linear equations:
$$\begin{align*} x + 2y - 3z &= 1 \ 2x + 3y + 4z &= 2 \ 3x + 4y + 5z &= 3 \end{align*}$$
We can represent this system using an augmented matrix:
$$\begin{pmatrix} 1 & 2 & -3 & 1 \ 2 & 3 & 4 & 2 \ 3 & 4 & 5 & 3 \end{pmatrix}$$
Now, we perform row operations to transform this matrix into RREF:
Subtract 2 times the first row from the second row.
Subtract 3 times the first row from the third row.
This results in the following matrix:
$$\begin{pmatrix} 1 & 2 & -3 & 1 \ 0 & -1 & 10 & 0 \ 0 & -2 & 14 & 0 \end{pmatrix}$$
Next, we eliminate the leading coefficient in the second column:
Multiply the second row by -1.
Add 2 times the second row to the third row.
We now have:
$$\begin{pmatrix} 1 & 2 & -3 & 1 \ 0 & 1 & -10 & 0 \ 0 & 0 & 2 & 0 \end{pmatrix}$$
Continue performing row operations until the matrix is in RREF:
Multiply the third row by 1/2.
Add 10 times the third row to the second row, and then subtract 3 times the third row from
the first row.
The resulting matrix is:
$$\begin{pmatrix} 1 & 2 & 0 & 1 \ 0 & 1 & 0 & -5 \ 0 & 0 & 1 & 0 \end{pmatrix}$$
From this RREF, we can easily read off the solution:
$$\begin{align*} x &= 1 \ y &= -5 \ z &= 0 \end{align*}$$
Matrix equations are a powerful tool in linear algebra, allowing us to solve systems of linear
equations in a compact and elegant way. In this chapter, we explore various methods to
solve matrix equations, with examples drawn from the video tutorials.
Row Operations and Gaussian Elimination
One of the most fundamental methods for solving matrix equations is through row operations
and Gaussian elimination. This process involves transforming the given matrix into reduced
row echelon form (RREF), from which we can easily read off the solution.
For instance, consider the following system of linear equations:
$$\begin{align*} x + 2y - 3z &= 1 \ 2x + 3y + 4z &= 2 \ 3x + 4y + 5z &= 3 \end{align*}$$
We can represent this system using an augmented matrix:
$$\begin{pmatrix} 1 & 2 & -3 & 1 \ 2 & 3 & 4 & 2 \ 3 & 4 & 5 & 3 \end{pmatrix}$$
Now, we perform row operations to transform this matrix into RREF:
Subtract 2 times the first row from the second row.
Subtract 3 times the first row from the third row.
This results in the following matrix:
$$\begin{pmatrix} 1 & 2 & -3 & 1 \ 0 & -1 & 10 & 0 \ 0 & -2 & 14 & 0 \end{pmatrix}$$
Next, we eliminate the leading coefficient in the second column:
Multiply the second row by -1.
Add 2 times the second row to the third row.
We now have:
$$\begin{pmatrix} 1 & 2 & -3 & 1 \ 0 & 1 & -10 & 0 \ 0 & 0 & 2 & 0 \end{pmatrix}$$
Continue performing row operations until the matrix is in RREF:
Multiply the third row by 1/2.
Add 10 times the third row to the second row, and then subtract 3 times the third row from
the first row.
The resulting matrix is:
$$\begin{pmatrix} 1 & 2 & 0 & 1 \ 0 & 1 & 0 & -5 \ 0 & 0 & 1 & 0 \end{pmatrix}$$
From this RREF, we can easily read off the solution:
$$\begin{align*} x &= 1 \ y &= -5 \ z &= 0 \end{align*}$$
