Consider the following statements in respect of square matrices A and B of same order : 1. If AB is a null matrix, then at least one of A and B is a null matrix. 2. If AB is an identity matrix, then BA = AB. Which of the above statements is/are correct?
2 only
Analyzing Matrix Properties: Null and Identity Matrices
We are asked to evaluate two statements about square matrices A and B of the same order.
Analyzing Statement 1: AB is a null matrix implies A or B is a null matrix
Statement 1 says: If AB is a null matrix, then at least one of A and B is a null matrix.
Let's consider an example of two non-null square matrices A and B whose product AB is a null matrix. This would be a counterexample to the statement.
Consider the following 2x2 matrices:
\[ A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \] \ \[ B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \]
Neither matrix A nor matrix B is a null matrix (a matrix where all entries are zero). Now let's calculate their product AB:
\[ AB = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} (0)(0) + (1)(0) & (0)(1) + (1)(0) \\ (0)(0) + (0)(0) & (0)(1) + (0)(0) \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]
The product AB is the null matrix. In this example, AB is a null matrix, but neither A nor B is a null matrix.
This counterexample shows that Statement 1 is false.
Analyzing Statement 2: AB is an identity matrix implies BA = AB
Statement 2 says: If AB is an identity matrix, then BA = AB.
If A and B are square matrices of the same order and AB is the identity matrix (denoted by I), this means that B is the inverse of A, and A is the inverse of B. This is written as \(B = A^{-1}\) or \(A = B^{-1}\).
A fundamental property of matrix inverses for square matrices is that if AB = I, then it is also true that BA = I. This property does not necessarily hold for non-square matrices, but it is true for square matrices of the same order.
Since AB = I and BA = I, it follows directly that AB = BA.
Therefore, Statement 2 is correct.
Summary of Analysis
- Statement 1: If AB is a null matrix, then at least one of A and B is a null matrix. (False)
- Statement 2: If AB is an identity matrix, then BA = AB. (True)
Based on the analysis, only Statement 2 is correct.
| Statement | Assessment | Reasoning |
|---|---|---|
| 1. If AB is a null matrix, then at least one of A and B is a null matrix. | Incorrect | Counterexample exists where AB is null, but A and B are non-null. |
| 2. If AB is an identity matrix, then BA = AB. | Correct | For square matrices, AB=I implies BA=I. Thus, BA = AB. |
Comparing our findings with the given options:
- 1. 1 only (Incorrect, Statement 1 is false)
- 2. 2 only (Correct, Statement 2 is true)
- 3. Both 1 and 2 (Incorrect, Statement 1 is false)
- 4. Neither 1 nor 2 (Incorrect, Statement 2 is true)
Revision Table: Matrix Properties Review
| Concept | Description | Important Note for Square Matrices |
|---|---|---|
| Null Matrix (O) | A matrix where all elements are zero. | If A and B are matrices, AB=O does NOT necessarily mean A=O or B=O (unlike scalar multiplication). |
| Identity Matrix (I) | A square matrix with ones on the main diagonal and zeros elsewhere. Acts like '1' in multiplication. | AI = IA = A for any matrix A of compatible size. |
| Matrix Inverse (\(A^{-1}\)) | For a square matrix A, its inverse \(A^{-1}\) satisfies \(AA^{-1} = A^{-1}A = I\). | If AB=I for square matrices A, B, then BA=I is automatically true. B is the inverse of A, and A is the inverse of B. |
| Commutativity | AB = BA. Matrix multiplication is generally NOT commutative (AB ≠ BA). | However, if AB=I for square matrices, then AB=BA=I, so they commute in this specific case. |
Additional Information: Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra. Here are some key points relevant to the question:
- Definition: The product AB of two matrices A and B is defined only if the number of columns in A is equal to the number of rows in B.
- Order of Product: If A is an \(m \times n\) matrix and B is an \(n \times p\) matrix, the product AB is an \(m \times p\) matrix. The element in the \(i\)-th row and \(j\)-th column of AB is the sum of the products of corresponding elements from the \(i\)-th row of A and the \(j\)-th column of B.
- Non-Commutativity: In general, matrix multiplication is not commutative, meaning AB ≠ BA, even if both products are defined and of the same order. For example, if A and B are square matrices of the same order, AB and BA are both defined and are of the same order, but they are usually not equal.
- Properties: Matrix multiplication is associative \((A(BC) = (AB)C)\) and distributive over matrix addition \((A(B+C) = AB+AC)\) and \(( (A+B)C = AC+BC)\).
- Zero Product: As seen in the analysis of Statement 1, it is possible for the product of two non-zero matrices to be the zero matrix (null matrix). This is different from multiplication of real numbers, where if \(xy=0\), then either \(x=0\) or \(y=0\).
Understanding these properties is crucial for working with matrices, especially when dealing with concepts like null matrices, identity matrices, and inverses.
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