If \( A \) is a square matrix of order 3 and \( |A| = -3 \), then the value of \( |2AA^T| \) is:
72
Calculating Determinant of \( |2AA^T| \) for a Square Matrix
The problem asks us to find the value of the determinant of the matrix \( 2AA^T \), given that \( A \) is a square matrix of order 3 and its determinant, \( |A| \), is equal to -3.
To solve this, we need to use several properties of determinants of matrices.
Key Determinant Properties
Let \( A \) and \( B \) be square matrices of the same order, and let \( k \) be a scalar.
- Determinant of a scalar multiple: If \( A \) is of order \( n \), then \( |kA| = k^n |A| \).
- Determinant of a product: \( |AB| = |A| |B| \).
- Determinant of a transpose: \( |A^T| = |A| \).
Step-by-Step Calculation of \( |2AA^T| \)
We are asked to find \( |2AA^T| \). Let's break this down using the properties.
First, consider the scalar multiple. The matrix inside the determinant is \( 2AA^T \). This can be seen as a scalar 2 multiplying the matrix \( AA^T \). Since \( A \) is a matrix of order 3, \( AA^T \) is also a matrix of order 3. Using the scalar multiple property \( |kA| = k^n |A| \) with \( k=2 \) and the matrix \( AA^T \) (which is of order \( n=3 \)):
Now, calculate \( 2^3 \):
Next, let's find \( |AA^T| \). Using the determinant of a product property \( |AB| = |A| |B| \) with \( B = A^T \):
Now, use the determinant of a transpose property \( |A^T| = |A| \):
Substitute this into the expression for \( |AA^T| \):
Finally, substitute this back into the expression for \( |2AA^T| \):
We are given that \( |A| = -3 \). Substitute this value into the equation:
Calculate \( (-3)^2 \):
Now, complete the calculation:
Thus, the value of \( |2AA^T| \) is 72.
Revision Table: Matrix Determinant Properties
| Property | Formula (A is order n, k is scalar) |
|---|---|
| Scalar Multiple | \( |kA| = k^n |A| \) |
| Product | \( |AB| = |A| |B| \) |
| Transpose | \( |A^T| = |A| \) |
Additional Information on Matrix Determinants and Order
The order of a square matrix refers to the number of rows (or columns) it has. A matrix of order 3 has 3 rows and 3 columns.
The determinant of a square matrix is a scalar value that can be computed from the elements of the matrix. It provides important information about the matrix, such as whether the matrix is invertible (a matrix is invertible if and only if its determinant is non-zero).
The properties used in this calculation are fundamental in matrix algebra and are often applied when dealing with determinants of matrix expressions involving products, transposes, or scalar multiples. Understanding these properties is crucial for solving problems related to matrix transformations and systems of linear equations.
For instance, the property \( |A^T| = |A| \) means that taking the transpose of a matrix does not change its determinant value. The property \( |AB| = |A||B| \) is very useful as it allows us to find the determinant of a product of matrices by simply multiplying their individual determinants.
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Important Questions from Determinants
An even number is the determinant of which of the following matrices?
(A) \(\begin{bmatrix} 1 & -1 \\ -1 & 5 \end{bmatrix}\)
(B) \(\begin{bmatrix} 13 & -1 \\ -1 & 15 \end{bmatrix}\)
(C) \(\begin{bmatrix} 16 & -1 \\ -11 & 15 \end{bmatrix}\)
(D) \(\begin{bmatrix} 6 & -12 \\ 11 & 15 \end{bmatrix}\)
Choose the correct answer from the options given below:
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Let X denote the sum of the numbers on the two cards drawn.
Then P(X > 3) is:
A random variable X has the following probability distribution:
X | -2 | -1 | 0 | 1 | 2 -------------------------------------------- P(X) | 0.2 | 0.1 | 0.3 | 0.2 | 0.2
The variance of X will be:
The angle between two lines whose direction ratios are proportional to \( (\sqrt{3} - 1) \), \( (-\sqrt{3} - 1) \), and -4 is:
