If \( B \) is a non-singular \( 4 \times 4 \) matrix and \( A \) is its adjoint such that \( |A| = 125 \), then \( |B| \) is:
5
Understanding the Matrix Adjoint and Determinant
This problem involves a non-singular matrix and its adjoint, and asks us to find the determinant of the original matrix given the determinant of its adjoint. We need to use a specific property that relates the determinant of a matrix to the determinant of its adjoint.
Key Property of Adjoint Matrix Determinant
For any square matrix \(B\) of size \(n \times n\), its adjoint is denoted as \(\text{adj}(B)\). If \(B\) is non-singular (meaning \(|B| \neq 0\)), there is a direct relationship between the determinant of the adjoint and the determinant of the matrix itself. The property is:
\( |\text{adj}(B)| = |B|^{n-1} \)
where \(n\) is the order (size) of the square matrix \(B\).
Applying the Property to the Given Problem
We are given that:
- \(B\) is a non-singular \(4 \times 4\) matrix. This means \(n = 4\).
- \(A\) is the adjoint of \(B\), so \(A = \text{adj}(B)\).
- The determinant of \(A\) is \(|A| = 125\). Since \(A = \text{adj}(B)\), we have \(|\text{adj}(B)| = 125\).
Now, we can use the property \( |\text{adj}(B)| = |B|^{n-1} \). Substituting the given values:
\( 125 = |B|^{4-1} \)
\( 125 = |B|^3 \)
Solving for the Determinant of B
We need to find the value of \(|B|\) such that its cube is 125. This is equivalent to finding the cube root of 125.
\( |B| = \sqrt[3]{125} \)
We know that \(5 \times 5 \times 5 = 25 \times 5 = 125\). Therefore, the cube root of 125 is 5.
\( |B| = 5 \)
Thus, the determinant of the matrix \(B\) is 5.
Let's verify this with the property:
If \(|B| = 5\) and \(n=4\), then \(|\text{adj}(B)| = |B|^{4-1} = 5^3 = 125\). This matches the given information.
Summary of Calculation
| Given Information | Matrix B is 4x4 (n=4) |
|---|---|
| Given Information | |adj(B)| = 125 |
| Property Used | |adj(B)| = |B|<sup>n-1</sup> |
| Substitution | 125 = |B|<sup>4-1</sup> |
| Simplification | 125 = |B|<sup>3</sup> |
| Solving for |B| | |B| = ∛125 |
| Result | |B| = 5 |
The determinant of \(B\) is 5.
Revision Table: Matrix Adjoint and Determinants
| Concept | Definition/Property |
|---|---|
| Adjoint of a Matrix (adj(B)) | The transpose of the cofactor matrix of B. |
| Determinant of a Matrix (|B|) | A scalar value that can be computed from the elements of a square matrix. It has various properties related to matrix invertibility and transformations. |
| Non-singular Matrix | A square matrix B for which |B| ≠ 0. Such a matrix has an inverse. |
| Relationship between B and adj(B) | \( B \cdot \text{adj}(B) = \text{adj}(B) \cdot B = |B| \cdot I \), where I is the identity matrix. |
| Determinant of Adjoint Property | For an \(n \times n\) non-singular matrix B, \( |\text{adj}(B)| = |B|^{n-1} \). |
Additional Information: Properties of Determinants
Determinants are fundamental in linear algebra and have several useful properties:
- The determinant of the identity matrix \(I_n\) is 1.
- The determinant of a transpose of a matrix \(B^T\) is equal to the determinant of the original matrix \(B\), i.e., \(|B^T| = |B|\).
- The determinant of the product of two square matrices \(B\) and \(C\) of the same size is the product of their determinants, i.e., \(|BC| = |B| \cdot |C|\).
- If a matrix has a row or column of zeros, its determinant is 0.
- If a matrix has two identical rows or columns, its determinant is 0.
- Multiplying a row or column by a scalar \(k\) multiplies the determinant by \(k\).
- Adding a multiple of one row/column to another row/column does not change the determinant.
- The determinant of an invertible matrix \(B\) is non-zero, and the determinant of its inverse \(B^{-1}\) is \( |B^{-1}| = 1/|B| \).
These properties are crucial for solving various problems involving matrices and their determinants.
Similar Questions
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 \( A \) is a square matrix of order 3 such that \( |2 \operatorname{adj} A| = 288 \), then the value of \( |A| \) 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:
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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:
