Matrix Equalities & Variables

Matrices can be equal if certain conditions are satisfied. Therefore, we can set up equations and solve for variables with two equal matrices. (Note: this is different from a Matrix Equation in which an entire matrix acts as a variable.)

Equal of Matrices

Two matrices are equal if and only if these matrices have the same dimensions and equal corresponding elements.

Which matrices below are equal?

All three matrices have the same dimensions

- 3 × 3 (3 rows and 3  columns)

All corresponding entries or elements are the same in matrix 1 and matrix 3.

- The middle most entry of matrix #2 is not the same as the corresponding entry in the other matrices. Therefore, matrix #2 does not equal either of the other ones.

Practice Problem

Problem 1

Which matrices below are equal?

Identify Complex Numbers on Graph

Matrices #4 and #5 are equal. They have the same dimensions and equal corresponding entries.

Problem 2

Which matrices below are equal?

Identify Complex Numbers on Graph

Matrices #8 and #9 are equal. They have the same dimensions and equal corresponding entries.

Problem 3

Which matrices below are equal?

Identify Complex Numbers on Graph

Matrix #10 and #11 are equal. Matrix #12 is ruled out because it does not have the same dimensions as the other two. It only has two columns

Solving for Variables in Matrices

If we know that two matrices are equal, we can find the value of variables in matrices. Since equal matrices have equal corresponding entries, we can set an unknown entry in one matrix equal to its corresponding partner in the other matrix.

To find the value of the variable y in the left hand matrix, we just set it equal to its corresponding entry in the right hand matrix.


y = 33
Problem 4

Solving for variables in equal matrices will not always be as easy as matching a variable with a corresponding number. Here's a slightly more challenging problem: What is the value of y?

3y = 33 (set corresponding entries equal)
3y ÷3 =33÷3 N
y=11

back to Matrices Home next to Multiplying Matrices

Ultimate Math Solver (Free)

Free Algebra Solver ... type anything in there!