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:
(A), (B) and (D) only
Understanding the Problem: Even Determinants of Matrices
The question asks us to identify which of the given 2x2 matrices have a determinant that is an even number. To solve this, we need to calculate the determinant for each matrix provided in the options and then determine if the calculated value is even or odd.
Calculating the Determinant of a 2x2 Matrix
For a 2x2 matrix given by \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), the determinant is calculated using the formula:
\( \text{Determinant} = ad - bc \)
We will apply this formula to each matrix.
Step-by-Step Determinant Calculation and Parity Check
Matrix (A) Determinant Calculation
The first matrix is \( \begin{bmatrix} 1 & -1 \\ -1 & 5 \end{bmatrix} \). Here, \(a=1\), \(b=-1\), \(c=-1\), and \(d=5\).
Determinant of (A) = \((1 \times 5) - (-1 \times -1)\)
= \(5 - (1)\)
= \(5 - 1\)
= \(4\)
The determinant is 4. Since 4 is divisible by 2, it is an even number.
Matrix (B) Determinant Calculation
The second matrix is \( \begin{bmatrix} 13 & -1 \\ -1 & 15 \end{bmatrix} \). Here, \(a=13\), \(b=-1\), \(c=-1\), and \(d=15\).
Determinant of (B) = \((13 \times 15) - (-1 \times -1)\)
= \(195 - (1)\)
= \(195 - 1\)
= \(194\)
The determinant is 194. Since 194 is divisible by 2 (\(194 = 2 \times 97\)), it is an even number.
Matrix (C) Determinant Calculation
The third matrix is \( \begin{bmatrix} 16 & -1 \\ -11 & 15 \end{bmatrix} \). Here, \(a=16\), \(b=-1\), \(c=-11\), and \(d=15\).
Determinant of (C) = \((16 \times 15) - (-1 \times -11)\)
= \(240 - (11)\)
= \(240 - 11\)
= \(229\)
The determinant is 229. Since 229 is not divisible by 2, it is an odd number.
Matrix (D) Determinant Calculation
The fourth matrix is \( \begin{bmatrix} 6 & -12 \\ 11 & 15 \end{bmatrix} \). Here, \(a=6\), \(b=-12\), \(c=11\), and \(d=15\).
Determinant of (D) = \((6 \times 15) - (-12 \times 11)\)
= \(90 - (-132)\)
= \(90 + 132\)
= \(222\)
The determinant is 222. Since 222 is divisible by 2 (\(222 = 2 \times 111\)), it is an even number.
Summary of Determinants and Parity
Let's summarize the results for each matrix:
| Matrix | Determinant Value | Parity (Even/Odd) |
|---|---|---|
| (A) | 4 | Even |
| (B) | 194 | Even |
| (C) | 229 | Odd |
| (D) | 222 | Even |
Based on our calculations, the matrices with an even determinant are (A), (B), and (D).
Conclusion
The matrices (A), (B), and (D) have even determinants. Therefore, the correct option should list (A), (B), and (D).
Revision Table: Key Concepts in Matrix Determinants
| Concept | Description | 2x2 Formula |
|---|---|---|
| Determinant | A scalar value computed from the elements of a square matrix. It provides important information about the matrix, such as invertibility. | For \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), determinant is \(ad - bc\) |
| Even Number | An integer that is exactly divisible by 2. | Examples: -4, 0, 2, 4, 194, 222 |
| Odd Number | An integer that is not exactly divisible by 2. | Examples: -3, -1, 1, 3, 5, 229 |
Additional Information: Properties of Determinants
Determinants are fundamental in linear algebra and have several key properties:
- A square matrix is invertible if and only if its determinant is non-zero.
- The determinant of a product of matrices is the product of their determinants: \( \text{det}(AB) = \text{det}(A) \text{det}(B) \).
- The determinant of the transpose of a matrix is equal to the determinant of the original matrix: \( \text{det}(A^T) = \text{det}(A) \).
- If a matrix has a row or a column of zeros, its determinant is zero.
- If a matrix has two identical rows or columns, its determinant is zero.
- Swapping two rows or columns of a matrix multiplies the determinant by -1.
- Multiplying a row or column by a scalar \(k\) multiplies the determinant by \(k\).
Understanding how to calculate determinants is crucial for solving systems of linear equations (using Cramer's rule), finding eigenvalues, and performing other matrix operations.
Important Questions from Determinants
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:
Given the determinant:
\[ \Delta = \begin{vmatrix} 1 & \cos x & 1 \\ -\cos x & 1 & \cos x \\ -1 & -\cos x & 1 \end{vmatrix} \]
Which of the following statements are correct?
(A) \( \Delta = 2(1 - \cos^2 x) \)
(B) \( \Delta = 2(2 - \sin^2 x) \)
(C) Minimum value of \( \Delta \) is 2
(D) Maximum value of \( \Delta \) is 4
Choose the correct answer from the options given below:
