Question 1:
Consider the matrices A =
[
2
3
4
1
]
[
2
4
3
1
] and B =
[
−
1
2
0
3
]
[
−1
0
2
3
]. Calculate the product AB if possible.
Solution 1:
To calculate the product AB, we multiply the elements of each row of matrix A with the corresponding elements of each column of matrix B and then sum the products. Here's how the calculation is done: =
[
2
3
4
1
]
[
−
1
2
0
3
]
=
[
(
2
⋅
−
1
+
3
⋅
0
)
(
2
⋅
2
+
3
⋅
3
)
(
4
⋅
−
1
+
1
⋅
0
)
(
4
⋅
2
+
1
⋅
3
)
]
=
[
−
2
13
−
4
11
]
AB=[
2
4
3
1
][
−1
0
2
3
]=[
(2⋅−1+3⋅0)
(4⋅−1+1⋅0)
(2⋅2+3⋅3)
(4⋅2+1⋅3)
]=[
−2
−4
13
11
]
Question 2:
Find the transpose of the matrix C =
[
5
−
2
7
0
3
1
]
[
5
0
−2
3
7
1
].
Solution 2:
To find the transpose of matrix C, we simply swap its rows with columns. Here's the calculation:
Transpose of C =
[
5
0
−
2
3
7
1
]
⎣
⎡
5
−2
7
0
3
1
⎦
⎤
Question 3:
If matrix D =
[
2
1
−
3
0
]
[
2
−3
1
0
], calculate its determinant.
Solution 3:
To calculate the determinant of matrix D, we use the formula det(D)=ad−bc, where a, b, c, and d are the elements of the matrix. Here's the calculation: ( )
=
(
2
⋅
0
)
−
(
−
3
⋅
1
)
=
0
+
3
=
3
det(D)=(2⋅0)−(−3⋅1)=0+3=3
