Equal Matrices
2
1
[3 −4]
=
2
1
[3 −4]
Lesson Objectives
You will be able to
▸ understand that two matrices are equal if they have the same order and the same corresponding
elements,
▸ find missing values that make two matrices equal by forming and solving equations.
Equality of Variables
In conventional algebra, two quantities are equal if they have the same value.
▸ For example, if we have 𝑥 = 5 and 𝑦 = 5, then we can say that the two quantities are equal and hence
write 𝑥 = 𝑦.
▸ Alternatively, if we have 𝑎 = 5 and 𝑏 = −10, then clearly these quantities are not equal and we would
write 𝑎 ≠ 𝑏.
Definition: Equality of Two Matrices
Consider two matrices, 𝐴 with order 𝑚 × 𝑛 and 𝐵 with order 𝑝 × 𝑞, described by their entries as follows:
𝐴 = (𝑎𝑖𝑗 ) ,
𝐵 = (𝑏𝑖𝑗 ) .
Then, we say that the two matrices are equal, that is, 𝐴 = 𝐵, if their dimensions are equal and the
corresponding entries are identical. In other words, the following two conditions must be satisfied:
𝑚 = 𝑝,
𝑎𝑖𝑗 = 𝑏𝑖𝑗 ,
𝑛 = 𝑞,
for all 𝑖, 𝑗.
Conversely, if 𝑚 ≠ 𝑝 or 𝑛 ≠ 𝑞, or there are any 𝑖 and 𝑗 such that 𝑎𝑖𝑗 ≠ 𝑏𝑖𝑗 , then the two matrices are not
equal; that is, 𝐴 ≠ 𝐵.
Example 1: Conditions for the Equality of Matrices
Given that
𝐴 = (3
3
3
3
3) ,
3
𝐵 = (3
3
3) ,
3
is it true that 𝐴 = 𝐵?
Answer
Matrices 𝐴 and 𝐵 can be written as
𝐴 = (𝑎𝑖𝑗 ) ,
𝐵 = (𝑏𝑖𝑗 ) .
Recall that in order for two matrices to be equal, both their dimensions and their entries must be equal. In
this case, we notice that some of the pairs of entries are equal. For instance, we have
𝑎11 = 𝑏11 = 3,
𝑎12 = 𝑏12 = 3,
𝑎21 = 𝑏21 = 3,
𝑎22 = 𝑏22 = 3.
Example 1 (Continued)
However, the orders of the matrices are not equal. Matrix 𝐴 has two rows and three columns, so it is a 2 × 3
matrix, while matrix 𝐵 has two rows and two columns, making it a 2 × 2 matrix. We highlight this extra
column below:
𝐴 = (3
3
3
3
3) .
3
As the orders of these matrices are not equivalent, it is not true that 𝐴 = 𝐵.
Example 2: Identifying Equality of Matrices
If
𝐴 = (−5
−7
3 ),
−3
𝐵 = (−5
−7
−3) ,
3
is it true that 𝐴 = 𝐵?
Answer
Recall that to check whether two matrices are equal, we have to confirm that they have the same order and
that 𝑎𝑖𝑗 = 𝑏𝑖𝑗 for all 𝑖 and 𝑗.
Both of these matrices have order 2 × 2, so to check for equality, we will have to check every entry. In the
matrices below, we have highlighted every entry in a different color to provide an easy comparison:
𝐴 = (−5 3 ) ,
−7 −3
𝐵 = (−5 −3) .
−7 3
Example 2 (Continued)
Comparing the top-left entries, we find that 𝑎11 = 𝑏11 = −5.
Comparing the bottom-left entries, we get 𝑎21 = 𝑏21 = −7.
However, examining the entries in the top right, we find that 𝑎12 = 3, while 𝑏12 = −3; hence, 𝑎12 ≠ 𝑏12 .
Similarly, in the bottom right, we can see that 𝑎22 = −3, while 𝑏22 = 3, so 𝑎22 ≠ 𝑏22 .
As these matrices do not satisfy the condition of 𝑎𝑖𝑗 = 𝑏𝑖𝑗 for all 𝑖, 𝑗, they are not equal.
Example 3: Solving Equations Using Matrix Equality
Given that
3𝑥 − 3 −3
0
=
( −10
)
(
𝑦−1
−10
−3
5𝑦 − 5) ,
find the values of 𝑥 and 𝑦.
Answer
Recall that to check whether two matrices are equal, we have to confirm that they have the same order and
that 𝑎𝑖𝑗 = 𝑏𝑖𝑗 for all 𝑖 and 𝑗.
We begin by highlighting all of the entries that we must compare:
3𝑥 − 3
( −10
−3
0
−3
=
)
(
𝑦−1
−10 5𝑦 − 5) .
There are two pairs of entries that are clearly equal in both matrices, namely, 𝑎12 = 𝑏12 = −3 and
𝑎21 = 𝑏21 = −10.
Example 3 (Continued)
To ensure that these matrices are equal, we set 𝑎11 = 𝑏11 , which implies that 3𝑥 − 3 = 0, giving 𝑥 = 1.
We now set 𝑎22 = 𝑏22 , which gives 𝑦 − 1 = 5𝑦 − 5, and hence 𝑦 = 1.
Thus, the final matrix is
( 0
−10
and 𝑥 = 1 and 𝑦 = 1.
−3) ,
0
Example 4: Solving Equations Using Matrix Equality
Find the values of 𝑥 and 𝑦, given the following:
2
10𝑥
+ 10 2) = ( 20
(
2𝑦 + 9
−3
9
2
9) .
Answer
Recall that to check whether two matrices are equal, we have to confirm that they have the same order and
that 𝑎𝑖𝑗 = 𝑏𝑖𝑗 for all 𝑖 and 𝑗.
We highlight each pair of entries as shown:
Clearly, we already have 𝑎12
entries.
2
2
10𝑥
+ 10 2) = ( 20
(
2𝑦 + 9 9) .
−3
9
= 𝑏12 = 2 and 𝑎22 = 𝑏22 = 9, so there are no further checks needed for these
Example 4 (Continued)
By setting 𝑎11 = 𝑏11 , we obtain the equation 10𝑥2 + 10 = 20.
Solving this for 𝑥, we get
10𝑥2 = 10
𝑥2 = 1
𝑥 = ±1.
By setting 𝑎21 = 𝑏21 , we have −3 = 2𝑦 + 9, implying that 𝑦 = −6.
In summary, 𝑥 = ±1 and 𝑦 = −6.
Example 5: Solving Equations Using Matrix Equality
Given that
𝑎+𝑏
(
𝑎+𝑏+𝑐
determine the values of 𝑎, 𝑏, 𝑐, and 𝑑.
𝑎−𝑏
) = (−3
−5
𝑎 − 7𝑏 − 𝑑
−17) ,
−64
Answer
Recall that for two matrices of the same order, they are only equal if all their corresponding entries are
equal. Thus, we can make a comparison for each entry:
𝑎+𝑏
(
𝑎+𝑏+𝑐
𝑎−𝑏
) = (−3 −17) ,
−5 −64
𝑎 − 7𝑏 − 𝑑
Example 5 (Continued)
which gives the following system of linear equations:
𝑎 + 𝑏 = −3,
𝑎 − 𝑏 = −17,
𝑎 + 𝑏 + 𝑐 = −5,
𝑎 − 7𝑏 − 𝑑 = −64.
We notice that the first two equations, 𝑎 + 𝑏 = −3 and 𝑎 − 𝑏 = −17, can be added together to eliminate the
𝑏 term.
This gives us
2𝑎 = −20
𝑎 = −10.
Example 5 (Continued)
Substituting this value of 𝑎 into the first equation (although the second can also be used), we get
−10 + 𝑏 = −3
𝑏 = 7.
Now that we have found 𝑎 and 𝑏, we notice that we can solve 𝑎 + 𝑏 + 𝑐 = −5 for 𝑐. This gives us
−10 + 7 + 𝑐 = −5
𝑐 = −2.
Finally, we can find 𝑑 from the last equation, 𝑎 − 7𝑏 − 𝑑 = −64:
−10 − 7 × 7 − 𝑑 = −64
−𝑑 = −5
𝑑 = 5.
In summary, we get 𝑎 = −10, 𝑏 = 7, 𝑐 = −2, and 𝑑 = 5.
Key Points
▸ Consider two matrices, 𝐴 with order 𝑚 × 𝑛 and 𝐵 with order 𝑝 × 𝑞, described by their entries as follows:
𝐴 = (𝑎𝑖𝑗 ) ,
𝐵 = (𝑏𝑖𝑗 ) .
Then, 𝐴 = 𝐵 if and only if
𝑚 = 𝑝,
𝑎𝑖𝑗 = 𝑏𝑖𝑗 ,
𝑛 = 𝑞,
for all 𝑖, 𝑗.
▸ Matrix equality is a strict condition. If there are any 𝑖 and 𝑗 such that 𝑎𝑖𝑗 ≠ 𝑏𝑖𝑗 , then 𝐴 ≠ 𝐵.
▸ Additionally, if 𝐴 and 𝐵 are of different orders, then 𝐴 ≠ 𝐵.
▸ We can find missing values that make two matrices equal by forming and solving equations.