If A = \(\left[\begin{array}{cc}2 & −3 \\3 & 5\end{array}\right]\), then which of the following statements are correct? A. A is a square matrix B. A−1 exists C. A is a symmetric matrix D. |A| = 19 E. A is a null matrix Choose the correct answer from the options given below.
A, B, D only
Analyzing Matrix Properties
We are given a matrix A and several statements about its properties. We need to evaluate each statement to determine which ones are correct. The given matrix is:
\(A = \left[\begin{array}{cc}2 & -3 \\3 & 5\end{array}\right]\)
Evaluating Statement A: A is a square matrix
A square matrix is a matrix that has the same number of rows as columns. Let's look at the dimensions of matrix A. Matrix A has 2 rows and 2 columns.
- Number of rows = 2
- Number of columns = 2
Since the number of rows equals the number of columns, matrix A is a square matrix. Therefore, statement A is correct.
Evaluating Statement B: A⁻¹ exists
The inverse of a matrix, denoted as A⁻¹, exists if and only if the determinant of the matrix is non-zero. That is, \(|A| \neq 0\). To evaluate this statement, we first need to calculate the determinant of A. This relates to statement D.
Evaluating Statement C: A is a symmetric matrix
A symmetric matrix is a square matrix that is equal to its transpose. The transpose of a matrix A, denoted as Aᵀ, is obtained by interchanging its rows and columns. For matrix A = \(\left[\begin{array}{cc}a & b \\c & d\end{array}\right]\), the transpose is Aᵀ = \(\left[\begin{array}{cc}a & c \\b & d\end{array}\right]\).
Let's find the transpose of matrix A:
\(A = \left[\begin{array}{cc}2 & -3 \\3 & 5\end{array}\right]\)
\(A^T = \left[\begin{array}{cc}2 & 3 \\-3 & 5\end{array}\right]\)
For A to be symmetric, A must be equal to Aᵀ. Comparing the elements:
- The element in the first row, second column of A is -3.
- The element in the first row, second column of Aᵀ is 3.
Since -3 is not equal to 3, A is not equal to Aᵀ. Therefore, statement C is incorrect. A is not a symmetric matrix.
Evaluating Statement D: |A| = 19
The determinant of a 2x2 matrix \(\left[\begin{array}{cc}a & b \\c & d\end{array}\right]\) is calculated as \(ad - bc\).
For matrix A = \(\left[\begin{array}{cc}2 & -3 \\3 & 5\end{array}\right]\), we have \(a=2\), \(b=-3\), \(c=3\), and \(d=5\). Let's calculate the determinant |A|:
\(|A| = (2)(5) - (-3)(3)\)
\(|A| = 10 - (-9)\)
\(|A| = 10 + 9\)
\(|A| = 19\)
So, the determinant of A is indeed 19. Therefore, statement D is correct.
Revisiting Statement B: A⁻¹ exists
As determined in the evaluation of statement D, the determinant of A is |A| = 19. Since the determinant is non-zero (19 ≠ 0), the inverse of matrix A exists. Therefore, statement B is correct.
Evaluating Statement E: A is a null matrix
A null matrix (or zero matrix) is a matrix in which all elements are zero. For example, a 2x2 null matrix is \(\left[\begin{array}{cc}0 & 0 \\0 & 0\end{array}\right]\). Matrix A is \(\left[\begin{array}{cc}2 & -3 \\3 & 5\end{array}\right]\). Since the elements of A are not all zero, A is not a null matrix. Therefore, statement E is incorrect.
Summary of Correct Statements
Based on our analysis:
- Statement A: A is a square matrix (Correct)
- Statement B: A⁻¹ exists (Correct, because |A| ≠ 0)
- Statement C: A is a symmetric matrix (Incorrect)
- Statement D: |A| = 19 (Correct)
- Statement E: A is a null matrix (Incorrect)
The correct statements are A, B, and D.
| Statement | Evaluation | Correctness |
|---|---|---|
| A. A is a square matrix | Matrix A is 2x2 (rows = columns) | Correct |
| B. A⁻¹ exists | Determinant \(|A|=19 \neq 0\) | Correct |
| C. A is a symmetric matrix | \(A \neq A^T\) | Incorrect |
| D. |A| = 19 | \((2)(5) - (-3)(3) = 19\) | Correct |
| E. A is a null matrix | Elements are not all zero | Incorrect |
The option that lists A, B, and D only is the correct answer.
Revision Table: Key Matrix Definitions
| Term | Definition | Property |
|---|---|---|
| Square Matrix | A matrix with an equal number of rows and columns. | Required for determinant, inverse (if non-singular), symmetric, skew-symmetric matrices. |
| Inverse Matrix (A⁻¹) | A matrix B such that AB = BA = I (Identity matrix). | Exists only if the matrix is square and its determinant is non-zero. |
| Symmetric Matrix | A square matrix A where A = Aᵀ (transpose of A). | Elements \(a_{ij} = a_{ji}\) for all i, j. |
| Determinant (|A|) | A scalar value calculated from the elements of a square matrix. | Indicates properties like invertibility. For 2x2 \(\left[\begin{array}{cc}a & b \\c & d\end{array}\right]\), determinant is \(ad-bc\). |
| Null Matrix | A matrix where all elements are zero. | Represented by 0. Adding it to a matrix doesn't change the matrix. |
Additional Information: Matrix Properties and Determinants
Understanding matrix properties is fundamental in linear algebra. The determinant of a matrix is a single number that encodes many of the matrix's properties.
- Determinant and Invertibility: A square matrix is invertible (non-singular) if and only if its determinant is non-zero. If the determinant is zero, the matrix is singular and has no inverse.
- Determinant Calculation for Larger Matrices: For matrices larger than 2x2, the determinant calculation involves cofactor expansion or row reduction methods. For a 3x3 matrix, for example, it's a sum of 3 terms, each involving a 2x2 determinant.
- Properties of Symmetric Matrices: Symmetric matrices have many special properties. For instance, their eigenvalues are always real. They play a role in various areas like quadratic forms and optimization.
- Matrix Operations: Besides finding the determinant or transpose, common matrix operations include addition, subtraction, and multiplication (when dimensions allow).
The given problem covers several basic but important concepts related to matrix properties, essential for solving problems in linear algebra.
Important Questions from Matrix Algebra
Consider the system of equations: x + y = 2 and 2x + 2y = 5. This system has
The standard ordered basis of R 3 is {e 1, e 2, e 3} Let T : R 3 → R 3 be the linear transformation such that T(e 1) = 7e 1- 5e 3, T (e 2) = -2e 2+ 9e 3, T(e 3) = e 1+ e 2+ e 3. The standard matrix of T is:
The system of equations
x + y + z = 6;
x + 4y + 6z = 20;
x + 4y + λz = μ
has NO solution for values of λ and μ given by
What is the transformation matrix M that transforms a square in the xy-plane defined by (1, 1) T, (-1, 1) T, (-1, -1) T and (1, -1) T to a parallelogram whose corresponding vertices are (2, 1) T, (0, 1) T, (-2, -1) T and (0, -1) T?
- If A = \( \left[\begin{array}{cc}0 & 1 \\ −1 & 0\end{array}\right]\) and (aI 2 + bA)2 = A, then
