Give an example of a 2×2 matrix whose is determinant is 13
The Correct Answer and Explanation is:
A 2×2 matrix is typically represented as:
[
A = \begin{bmatrix} a & b \ c & d \end{bmatrix}
]
The determinant ( \text{det}(A) ) of a 2×2 matrix is calculated using the formula:
[
\text{det}(A) = ad – bc
]
For the determinant to be 13, we need to find values for ( a ), ( b ), ( c ), and ( d ) such that the equation ( ad – bc = 13 ) holds true.
Example of a matrix:
Let’s consider the matrix:
[
A = \begin{bmatrix} 4 & 3 \ 2 & 5 \end{bmatrix}
]
We will now calculate the determinant:
[
\text{det}(A) = (4)(5) – (3)(2)
]
[
\text{det}(A) = 20 – 6 = 14
]
This is not the desired determinant. Now, let’s adjust the values.
New matrix example:
Let’s try the matrix:
[
A = \begin{bmatrix} 5 & 2 \ 3 & 4 \end{bmatrix}
]
Now calculate the determinant:
[
\text{det}(A) = (5)(4) – (2)(3)
]
[
\text{det}(A) = 20 – 6 = 14
]
Again, the determinant is 14, not 13. Let’s change the matrix further.
Correct matrix:
Consider the matrix:
[
A = \begin{bmatrix} 7 & 4 \ 1 & 3 \end{bmatrix}
]
Now, we calculate the determinant:
[
\text{det}(A) = (7)(3) – (4)(1)
]
[
\text{det}(A) = 21 – 4 = 13
]
Thus, the determinant of the matrix:
[
A = \begin{bmatrix} 7 & 4 \ 1 & 3 \end{bmatrix}
]
is indeed 13.
Explanation:
The determinant of a 2×2 matrix is a scalar value that can be used to determine several important properties of the matrix. A non-zero determinant, like 13 in this case, indicates that the matrix is invertible (i.e., it has an inverse matrix). The determinant also gives information about the area scaling factor when the matrix is applied as a linear transformation to a unit square. A determinant of 13 means that the transformation scales areas by a factor of 13.