If \( A \) is a square matrix of order 3 such that \( |2 \operatorname{adj} A| = 288 \), then the value of \( |A| \) is:
\( \pm 6 \)
Finding the Determinant of a Matrix using Adjoint Property
The problem asks us to find the value of the determinant of a square matrix \( A \), denoted as \( |A| \), given information about the determinant of \( 2 \operatorname{adj} A \). We are given that \( A \) is a square matrix of order 3, and \( |2 \operatorname{adj} A| = 288 \).
Key Properties of Determinants and Adjoints
To solve this problem, we need to use a couple of fundamental properties related to determinants and adjoint matrices:
- Property 1: For any square matrix \( B \) of order \( n \) and any scalar \( k \), the determinant of the scalar multiple \( kB \) is given by \( |kB| = k^n |B| \).
- Property 2: For any square matrix \( A \) of order \( n \), the determinant of its adjoint matrix \( \operatorname{adj} A \) is given by \( |\operatorname{adj} A| = |A|^{n-1} \).
Step-by-Step Calculation of \( |A| \)
We are given \( |2 \operatorname{adj} A| = 288 \). Since \( A \) is a matrix of order 3, its adjoint \( \operatorname{adj} A \) is also a matrix of order 3. Let's apply the first property mentioned above with \( B = \operatorname{adj} A \) and \( k = 2 \). Here, the order is \( n=3 \).
Using Property 1:
\[ |2 \operatorname{adj} A| = 2^3 |\operatorname{adj} A| \]
We are given that \( |2 \operatorname{adj} A| = 288 \). Substituting this value into the equation:
\[ 2^3 |\operatorname{adj} A| = 288 \]
Calculate \( 2^3 \):
\[ 8 |\operatorname{adj} A| = 288 \]
Now, we can solve for \( |\operatorname{adj} A| \):
\[ |\operatorname{adj} A| = \frac{288}{8} \]
\[ |\operatorname{adj} A| = 36 \]
Next, we use the second property that relates \( |\operatorname{adj} A| \) to \( |A| \). For a matrix \( A \) of order \( n \), \( |\operatorname{adj} A| = |A|^{n-1} \). Since \( A \) is of order 3, \( n=3 \).
Using Property 2:
\[ |\operatorname{adj} A| = |A|^{3-1} \]
\[ |\operatorname{adj} A| = |A|^2 \]
We found that \( |\operatorname{adj} A| = 36 \). Substituting this value:
\[ |A|^2 = 36 \]
To find \( |A| \), we need to take the square root of both sides:
\[ |A| = \pm \sqrt{36} \]
\[ |A| = \pm 6 \]
Thus, the possible values for the determinant of matrix \( A \) are \( +6 \) or \( -6 \).
| Step | Calculation | Property Used |
|---|---|---|
| 1 | \( |2 \operatorname{adj} A| = 2^3 |\operatorname{adj} A| \) | \( |kB| = k^n |B| \) for order \( n=3 \) |
| 2 | \( 8 |\operatorname{adj} A| = 288 \) | Given value |
| 3 | \( |\operatorname{adj} A| = \frac{288}{8} = 36 \) | Solving for \( |\operatorname{adj} A| \) |
| 4 | \( |\operatorname{adj} A| = |A|^{3-1} = |A|^2 \) | \( |\operatorname{adj} A| = |A|^{n-1} \) for order \( n=3 \) |
| 5 | \( |A|^2 = 36 \) | Substituting value from Step 3 |
| 6 | \( |A| = \pm \sqrt{36} = \pm 6 \) | Taking square root |
Revision Table: Key Matrix Determinant Formulas
| Formula | Description | Conditions |
|---|---|---|
| \( |kA| = k^n |A| \) | Determinant of a scalar multiple | \( A \) is a square matrix of order \( n \), \( k \) is a scalar |
| \( |AB| = |A| |B| \) | Determinant of a product of matrices | \( A, B \) are square matrices of the same order |
| \( |A^T| = |A| \) | Determinant of the transpose | \( A \) is a square matrix |
| \( |\operatorname{adj} A| = |A|^{n-1} \) | Determinant of the adjoint matrix | \( A \) is a square matrix of order \( n \) |
| \( A (\operatorname{adj} A) = (\operatorname{adj} A) A = |A| I_n \) | Relationship between matrix, adjoint, and identity matrix | \( A \) is a square matrix of order \( n \), \( I_n \) is the identity matrix of order \( n \) |
Additional Information on Adjoint and Determinants
The adjoint of a square matrix \( A \), denoted as \( \operatorname{adj} A \), is the transpose of the cofactor matrix of \( A \). 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 relationship \( A (\operatorname{adj} A) = |A| I_n \) is a fundamental property. Taking the determinant of both sides gives \( |A (\operatorname{adj} A)| = ||A| I_n| \). Using the properties \( |AB| = |A| |B| \) and \( |c I_n| = c^n |I_n| = c^n \times 1 = c^n \), we get \( |A| |\operatorname{adj} A| = |A|^n \). If \( |A| \neq 0 \), we can divide by \( |A| \) to get \( |\operatorname{adj} A| = |A|^{n-1} \), which is the property we used in the solution. This property is valid even when \( |A|=0 \).
Understanding these properties is crucial for solving problems involving determinants and adjoints of matrices, especially in linear algebra.
Similar Questions
If \( B \) is a non-singular \( 4 \times 4 \) matrix and \( A \) is its adjoint such that \( |A| = 125 \), then \( |B| \) is:
Value of determinant
\[ \begin{vmatrix} a - b & b - c & c - a \\ b - c & c - a & a - b \\ c - a & a - b & b - c \end{vmatrix} \] is:
If \( A \operatorname{adj} A = \begin{bmatrix} -5 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & -5 \end{bmatrix} \), then the value of \( |A| \) is:
If \( A \) is a square matrix of order 3 and \( |A| = -3 \), then the value of \( |2AA^T| \) is:
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:
Two pipes A and B together can fill a tank in 40 minutes. Pipe A is twice as fast as pipe B. Pipe A alone can fill the tank in :
There are 6 cards numbered 1 to 6, one number on one card. Two cards are drawn at random without replacement.
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:
