12-1 Adding and Subtracting Matrices
Quick Review
To perform matrix addition or subtraction, add or subtract the corresponding elements in the matrices.
Two matrices are equal matrices when they have the same dimensions and corresponding elements are equal. This principle is used to solve a matrix equation.
Example
If
A
=
[
2
1
−
2
1
4
3
−
2
−
1
5
]
eh equals . matrix with 3 rows and 3 columns , row1 column 1 , 2 , column 2 1 , column 3 negative 2 , row2 column 1 , 1 , column 2 4 , column 3 3 , row3 column 1 , negative 2 , column 2 negative 1 , column 3 5 , end matrix and
B
=
[
1
−
2
4
−
3
−
2
1
0
0
5
]
,
b equals . matrix with 3 rows and 3 columns , row1 column 1 , 1 , column 2 negative 2 , column 3 4 , row2 column 1 , negative 3 , column 2 negative 2 , column 3 1 , row3 column 1 , 0 , column 2 0 , column 3 5 , end matrix . comma what is A + B?
A
+
B
=
[
2
+
1
1
+
(
−
2
)
−
2
+
4
1
+
(
−
3
)
4
+
(
−
2
)
3
+
1
−
2
+
0
−
1
+
0
5
+
5
]
=
[
3
−
1
2
−
2
2
4
−
2
−
1
10
]
table with 2 rows and 2 columns , row1 column 1 , eh plus b , column 2 equals . matrix with 3 rows and 3 columns , row1 column 1 , 2 plus 1 , column 2 1 plus . open , negative 2 , close , column 3 negative 2 plus 4 , row2 column 1 , 1 plus . open , negative 3 , close , column 2 4 plus . open , negative 2 , close , column 3 3 plus 1 , row3 column 1 , negative 2 plus 0 , column 2 negative 1 plus 0 , column 3 5 plus 5 , end matrix , row2 column 1 , , column 2 equals . matrix with 3 rows and 3 columns , row1 column 1 , 3 , column 2 negative 1 , column 3 2 , row2 column 1 , negative 2 , column 2 2 , column 3 4 , row3 column 1 , negative 2 , column 2 negative 1 , column 3 10 , end matrix , end table
Exercises
Find each sum or difference.
-
[
1
2
−
5
3
−
2
1
]
+
[
−
2
7
−
3
1
2
5
]
. matrix with 2 rows and 3 columns , row1 column 1 , 1 , column 2 2 , column 3 negative 5 , row2 column 1 , 3 , column 2 negative 2 , column 3 1 , end matrix . plus . matrix with 2 rows and 3 columns , row1 column 1 , negative 2 , column 2 7 , column 3 negative 3 , row2 column 1 , 1 , column 2 2 , column 3 5 , end matrix
-
[
0
2
−
4
−
1
]
−
[
−
5
6
−
9
−
1
]
. matrix with 2 rows and 2 columns , row1 column 1 , 0 , column 2 2 , row2 column 1 , negative 4 , column 2 negative 1 , end matrix . minus . matrix with 2 rows and 2 columns , row1 column 1 , negative 5 , column 2 6 , row2 column 1 , negative 9 , column 2 negative 1 , end matrix
Solve each matrix equation.
-
[
2
−
6
8
]
+
[
−
1
−
2
4
]
=
X
left bracket . table with 1 row and 3 columns , row1 column 1 , 2 , column 2 negative 6 , column 3 8 , end table . right bracket plus left bracket . table with 1 row and 3 columns , row1 column 1 , negative 1 , column 2 negative 2 , column 3 4 , end table . right bracket equals x
-
[
7
−
1
0
8
]
+
X
=
[
4
9
−
3
11
]
. matrix with 2 rows and 2 columns , row1 column 1 , 7 , column 2 negative 1 , row2 column 1 , 0 , column 2 8 , end matrix . plus x equals . matrix with 2 rows and 2 columns , row1 column 1 , 4 , column 2 9 , row2 column 1 , negative 3 , column 2 11 , end matrix
Find the value of each variable.
-
[
x
−
5
9
4
t
+
2
]
=
[
−
7
w
+
1
8
−
r
1
]
. matrix with 2 rows and 2 columns , row1 column 1 , x minus 5 , column 2 9 , row2 column 1 , 4 , column 2 t plus 2 , end matrix . equals . matrix with 2 rows and 2 columns , row1 column 1 , negative 7 , column 2 w plus 1 , row2 column 1 , 8 minus r , column 2 1 , end matrix
-
[
−
4
+
t
2
y
r
w
+
5
]
=
[
2
t
11
−
2
r
+
12
9
]
. matrix with 2 rows and 2 columns , row1 column 1 , negative 4 plus t , column 2 2 y , row2 column 1 , r , column 2 w plus 5 , end matrix . equals . matrix with 2 rows and 2 columns , row1 column 1 , 2 t , column 2 11 , row2 column 1 , negative 2 r plus 12 , column 2 9 , end matrix
12-2 Matrix Multiplication
Quick Review
To obtain the product of a matrix and a scalar, multiply each matrix element by the scalar. Matrix multiplication uses both multiplication and addition. The element in the ith row and the jth column of the product of two matrices is the sum of the products of each element of the ith row of the first matrix and the corresponding element of the jth column of the second matrix. The first matrix must have the same number of columns as the second has rows.
Example
If
A
=
[
1
−
3
−
2
0
]
eh equals . matrix with 2 rows and 2 columns , row1 column 1 , 1 , column 2 negative 3 , row2 column 1 , negative 2 , column 2 0 , end matrix and
B
=
[
1
4
0
2
]
,
b equals . matrix with 2 rows and 2 columns , row1 column 1 , 1 , column 2 4 , row2 column 1 , 0 , column 2 2 , end matrix . comma what is AB?
A
B
=
[
(
1
)
(
1
)
+
(
−
3
)
(
0
)
(
1
)
(
4
)
+
(
−
3
)
(
2
)
(
−
2
)
(
1
)
+
(
0
)
(
0
)
(
−
2
)
(
4
)
+
(
0
)
(
2
)
]
=
[
1
−
2
−
2
−
8
]
table with 2 rows and 2 columns , row1 column 1 , eh b , column 2 equals . matrix with 2 rows and 2 columns , row1 column 1 , open 1 close . open 1 close , plus . open , negative 3 , close . open 0 close , column 2 open 1 close . open 4 close , plus . open , negative 3 , close . open 2 close , row2 column 1 , open , negative 2 , close . open 1 close , plus , open 0 close . open 0 close , column 2 open , negative 2 , close . open 4 close , plus , open 0 close . open 2 close , end matrix , row2 column 1 , , column 2 equals . matrix with 2 rows and 2 columns , row1 column 1 , 1 , column 2 negative 2 , row2 column 1 , negative 2 , column 2 negative 8 , end matrix , end table
Exercises
Use matrices A, B, C, and D to find each scalar product and sum, or difference, if possible. If an operation is not defined, label it undefined.
-
A
=
[
6
1
0
8
−
4
3
7
11
]
eh equals . matrix with 2 rows and 4 columns , row1 column 1 , 6 , column 2 1 , column 3 0 , column 4 8 , row2 column 1 , negative 4 , column 2 3 , column 3 7 , column 4 11 , end matrix
-
B
=
[
1
3
−
2
4
]
b equals . matrix with 2 rows and 2 columns , row1 column 1 , 1 , column 2 3 , row2 column 1 , negative 2 , column 2 4 , end matrix
-
C
=
[
−
2
1
4
0
2
2
1
1
]
c equals . matrix with 4 rows and 2 columns , row1 column 1 , negative 2 , column 2 1 , row2 column 1 , 4 , column 2 0 , row3 column 1 , 2 , column 2 2 , row4 column 1 , 1 , column 2 1 , end matrix
-
D
=
[
5
−
2
3
6
]
d equals . matrix with 2 rows and 2 columns , row1 column 1 , 5 , column 2 negative 2 , row2 column 1 , 3 , column 2 6 , end matrix
- 3A
-
B
−
2
A
b minus , 2 eh
-
AB
-
BA
-
A
C
−
B
D
eh c minus b d
-
4
B
−
3
D
4 b minus 3 d
