If A is a square matrix of order 4 and |A|= 4, then |2A| will be:
64
Calculating Determinant of a Scaled Matrix
The question asks us to find the determinant of the matrix 2A, given that A is a square matrix of order 4 and its determinant, |A|, is 4.
To solve this, we need to use a fundamental property of determinants related to scalar multiplication of a matrix.
Property of Determinants with Scalar Multiplication
For any square matrix A of order \(n\) and any scalar \(k\), the determinant of the matrix \(kA\) is given by the formula:
\(|kA| = k^n |A|\)
Applying the Property to the Problem
In this specific problem:
- The matrix A is a square matrix.
- The order of the matrix, \(n\), is 4.
- The scalar multiplier, \(k\), is 2.
- The determinant of matrix A, \(|A|\), is given as 4.
We need to find \(|2A|\).
Using the formula \(|kA| = k^n |A|\), we substitute the values:
\(|2A| = 2^4 |A|\)
Calculation
First, calculate \(2^4\):
\(2^4 = 2 \times 2 \times 2 \times 2 = 16\)
Now, substitute this value and the given value of \(|A|\) into the equation:
\(|2A| = 16 \times 4\)
\(|2A| = 64\)
Thus, the determinant of 2A is 64.
Step-by-Step Solution
- Identify the given information: Matrix A is order 4, \(|A| = 4\), scalar multiplier is 2.
- Recall the property of determinants for scalar multiplication: \(|kA| = k^n |A|\).
- Identify the order of the matrix \(n = 4\) and the scalar \(k = 2\).
- Substitute \(k\) and \(n\) into the formula: \(|2A| = 2^4 |A|\).
- Calculate \(2^4\): \(2^4 = 16\).
- Substitute the value of \(|A|\) into the equation: \(|2A| = 16 \times 4\).
- Perform the final multiplication: \(|2A| = 64\).
The value of |2A| is 64.
Revision Table: Key Matrix Determinant Properties
| Property | Description | Formula |
|---|---|---|
| Determinant of Transpose | The determinant of the transpose of a matrix is equal to the determinant of the original matrix. | \(|A^T| = |A|\) |
| Determinant of Product | The determinant of the product of two matrices is the product of their determinants. | \(|AB| = |A| |B|\) |
| Determinant of Inverse | The determinant of the inverse of an invertible matrix is the reciprocal of the determinant of the original matrix. | \(|A^{-1}| = \frac{1}{|A|}\) (if \(|A| \neq 0\)) |
| Determinant with Scalar Multiplication | The determinant of a scalar times a matrix is the scalar raised to the power of the matrix order times the determinant of the original matrix. | \(|kA| = k^n |A|\) (for order n matrix A) |
Additional Information on Matrix Determinants
The determinant of a square matrix is a scalar value that can be computed from the elements of the matrix. It has many uses in linear algebra, including:
- Determining if a matrix is invertible (a matrix is invertible if and only if its determinant is non-zero).
- Solving systems of linear equations using Cramer's rule.
- Finding the area (for 2x2 matrices) or volume (for 3x3 matrices) of geometric shapes transformed by the matrix.
- Finding eigenvalues of a matrix.
The property \(|kA| = k^n |A|\) is particularly important when dealing with transformations that involve scaling. If a matrix A represents a linear transformation, then 2A represents a transformation that scales lengths by a factor of 2 in all directions. For an n-dimensional space, this scaling affects the 'volume' (represented by the determinant) by a factor of \(2^n\).
64
Use the determinant property: \[ |kA| = k^n \cdot |A| \] where:
\( k = 2 \), \( n = 4 \), and \( |A| = 4 \)
\[ |2A| = 2^4 \cdot 4 = 16 \cdot 4 = 64 \]
Answer:
64
64
Step 1: Recall the property of determinants under scalar multiplication
For an \( n \times n \) matrix \( A \) and scalar \( k \): \[ |kA| = k^n |A| \]
Step 2: Apply to our case
Given:
- Matrix order \( n = 4 \)
- Scalar \( k = 2 \)
- Original determinant \( |A| = 4 \)
\[ |2A| = 2^4 \times |A| = 16 \times 4 = 64 \]
Step 3: Verify the calculation
For a 4×4 matrix:
- Multiplying by 2 affects all 4 rows
- Each multiplication by 2 contributes a factor of 2 to the determinant
- Total factor is \( 2 \times 2 \times 2 \times 2 = 16 \)
Final Answer:
The determinant of \( 2A \) is \[ \boxed{64} \].
Key Points:
- For an \( n \times n \) matrix, scalar multiplication by \( k \) scales the determinant by \( k^n \)
- This is because the determinant is a multilinear function of the rows/columns
- The result holds for any square matrix, not just 4×4 matrices
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:
