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If one of these two conditions is not met, then matrix is not equal to matrix (written as ).
Matrix equality is decided by two checks. The two matrices must have the same format, and then every entry at the same position must match. The examples below show one pair that passes both checks and two pairs that fail exactly one of them, followed by a case where the entries contain unknown variables.
Given two matrices:
Matrix and matrix are equal () because:
Both have an order of .
Corresponding elements have the same value:
, , , .
Given two matrices:
Matrix is not equal to matrix () because has dimensions , while has dimensions .
Given two matrices:
Although matrix and matrix have the same dimensions, , they are not equal (). Their entries in row , column differ: , whereas .
Given the following matrices:
If matrix is equal to matrix (), determine the values of , , and .
Solution:
Since , the corresponding elements must be equal:
| Position | Equation and solution |
|---|---|
| (1, 1) | |
| (1, 2) | (already equal) |
| (2, 1) | |
| (2, 2) |
The values are , , and .
Decide whether the first three statements are True or False.
Having the same dimensions is one condition that two equal matrices must satisfy.
Two unequal matrices always have different dimensions.
The two matrices below are equal.
For the following matrices, . Determine and .
Matrix is the identity matrix. For the following matrix , . Determine .
Calculate the value of that satisfies the following matrix equality:
Problem 4. Given :
From the equality of corresponding elements:
Thus, and .
Problem 5. The identity matrix is:
Given :
From the equality of corresponding elements, we obtain a system of equations:
Solve Equation 2 first:
Substitute Equation 2' into Equation 1:
Substitute the value of into Equation 2':
Then, the value of is:
Problem 6. Given the matrix equality:
From the equality of corresponding elements, we obtain a system of equations:
Solve the system of equations for and (equations 1 and 2):
Initial equations:
To eliminate , multiply equation (2) by :
Subtract equation (1) from equation (2'):
Substitute the value of into equation (1):
Solve the system of equations for and (equations 3 and 4):
Initial equations:
Subtract equation (3) from equation (4) to eliminate :
Substitute the value of into equation (3):
Then, the value of is:
Published: . Updated: .
