4.4 Solving Systems With
Matrix Equations
Objective: Use matrices to solve systems of linear
equations in mathematical and real-world
situations.
Standard: 2.8.11.1. Use matrices to organize and
manipulate data.
Solving a matrix equation of the form AX = B, where
X = x y , is similar to solving a linear equation in the form ax = b,
where a, b, and x are real numbers and a ≠ 0.
Real Numbers
ax = b
½ (ax) = ½(b)
(1/a• a)x = b/a
x = b/a
Matrices
AX = B
A-1(AX) = A-1B
(A-1A)X = A-1B
IX = A-1B
X = A-1B
Just as 1/a must exist in order to solve ax = b (where a ≠
o), A-1 must exist to solve AX = B.
CALCULATOR: A-1* B
Ex 1. A financial manager wants to invest $50, 000
for a client by putting some of the money in a lowrisk investment that earns 5% per year and some of
the money in a high-risk investment that earns 14%
per year.
A). How much money should be invested at each
interest rate to earn $5000 in interest per year?
X + Y = 50,000
.05X + .14Y = 5,000
B). How much money should the manager invest at
each interest rate to earn $4000 in interest per
year?
X + Y = 50,000
.05X + .14Y = 5,000
5%  $33,333.33
14%  $16,666.67
Ex 3. Refer to the system of equations at right. 2y – z = 4x - 3
a. Write the system as a matrix equation.
2x + 3z = y – 6
b. Solve the matrix equation.
3y – 1 = 2x + 2z
-4x + 2y – z = -3
 2x – y + 3z = -6
 -2x + 3y – 2z = 1


X = 1 Y = -1 and Z = -3
-3x + 4y = 3
* Ex 4. Solve: -6x + 8y = 18 , if possible, by using a
matrix equation. If not possible, classify the system.
9x - 3y = 27
* Ex 5. Solve: - 6x + 2y = -18 , if possible, by using a
matrix equation. If not possible, classify the system.
Writing Activities

12). The system at the right can
be represented by a matrix
equation. What will be the
dimensions of the coefficient
matrix? Explain.