Solving Systems of Linear Equations with Matrices
Say you have a system of four linear equations in four variables like the following:
x  y
4w  1
2 x  2 y  3 z  2 w  1
4 y  6z  w  4
2x  4 y  9z
6
You can use substitution in the same way you did in Math 90 when you solved systems of
three equations in three variables, but there is an easier way!
You can put the coefficients of the system into a matrix. A matrix is just an array of
numbers in rows and columns. The matrix below represents this system of equations.
This matrix has four rows and five columns, so we say the dimension
is 4 x 5. The first column has the coefficients of x from each
equation. Notice that the third entry down is a 0 since there is no x
1 1
0 4 1
in the third equation. The second column has the coefficients of y


from each equation. Notice that the second entry down is –2 since
2  2  3 2  1
0 4
6 1 4
the coefficient of y in the second equation is –2. The last column


has the constants from the right hand side of the equals sign. That
9 0 6
2 4
is why this column is separated from the rest of the matrix by a
horizontal line.
We want to put the matrix into the calculator to get the Reduced Row Echelon Form of the
matrix. The Reduced Row Echelon Form gives the solution of the system of equations.
To put the matrix into the calculator, press 2nd and x–1 for
MATRIX, then press the right arrow two times to select EDIT.
Notice that Matrix A is already selected, so you press ENTER .
If you want to edit a matrix into B, C, D, etc., use the down
arrow to select a different matrix.
The first information you put in is the dimension of the matrix.
Since this is a 4 x 5 matrix, 4 ENTER 5 ENTER .
You can now put in the entries of the matrix one row at a time.
Be sure you press ENTER after each entry.
Always check to make sure you put all the entries in correctly.
You can use the arrow keys to move around your matrix when you
finish putting in all the entries.
Next, press 2nd MODE to QUIT. Always QUIT after entering
a matrix.
2
To get the Reduced Row Echelon Form of the matrix, go back
into MATRIX 2nd and x–1 and select MATH by pressing the
right arrow key once.
Move down until you see rref( . Then press ENTER .
To save time, notice that the shortcut to rref( was B. When you
get to the MATRIX MATH menu, you can press ALPHA APPS
for the letter B.
Next, go back into MATRIX. Matrix A is already selected in
NAMES, so press ENTER . Press ENTER once more to get the
Reduced Row Echelon form of the matrix.
Notice that one of the numbers does not have a terminating
decimal expansion, so the numbers would be more precise if they
were expressed as fractions.
To get any answer expressed as a fraction, press MATH and
ENTER two times.
This matrix represents the system of equations:
x 1
2
y1
2
z 1
3
w0
So, this is the solution of the system of equations.