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.)
Two matrices are equal if and only if these matrices have the same dimensions and equal corresponding elements.
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.
Matrices #4 and #5 are equal. They have the same dimensions and equal corresponding entries.
Matrices #8 and #9 are equal. They have the same dimensions and equal corresponding entries.
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.
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 = 33Solving 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
y = 11
