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In the world of matrices, we often need to compare two or more matrices. One important concept in this comparison is matrix equality. Two matrices are said to be equal if they meet certain conditions.
Two matrices, let's say matrix and matrix , are said to be equal (written as ) if and only if both of the following conditions are met:
If one of these two conditions is not met, then matrix is not equal to matrix (written as ).
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 the order of matrix is , while the order of matrix is .
Given two matrices:
Although matrix and matrix have the same order (), they are not equal () because the element in the nd row and st column is not the same ( while ).
Given matrices and .
If matrix is equal to matrix (), determine the values of , , and .
Solution:
Since , the corresponding elements must be equal:
Thus, the values are , , and .
Answer the following questions with True or False.
Two matrices having the same order is one of the conditions for the two matrices to be equal.
Two different matrices always have different orders.
If given matrix and matrix , then matrix is equal to matrix .
Given matrices and . If , determine the values of and .
If matrix and is the identity matrix of order . If , determine the value of .
Calculate the value of that satisfies the following matrix equality:
True. Having the same order is the first condition for two matrices to be equal.
False. Two different matrices can have the same order, but their corresponding elements are different (see Example ).
False. Matrix has an order of while matrix has an order of . Since their orders are different, the two matrices are not equal.
Given :
The identity matrix of order is .
Given the matrix equality:
From the equality of corresponding elements:
Thus, and .
Given :
From the equality of corresponding elements, we obtain a system of equations:
Solve Equation first:
Substitute Equation ' into Equation :
Substitute the value of into Equation :
Then, the value of is:
From the equality of corresponding elements, we obtain a system of equations:
Solve the system of equations for and (equations and ):
Initial equations:
To eliminate , multiply equation () by :
Subtract equation () from equation ('):
Substitute the value of into equation ():
Solve the system of equations for and (equations and ):
Initial equations:
Subtract equation () from equation () to eliminate :
Substitute the value of into equation ():
Then, the value of is:
