Let A be a non-singular matrix and B = adj A . Which of the following statements is/are correct? 1. AB = BA 2. AB is a scalar matrix 3. AB can be a null matrix Select the correct answer using the code given below:
Understanding Non-Singular Matrices and Adjoints
This question asks about the properties of the product of a non-singular matrix \(A\) and its adjoint, \(B = \text{adj } A\). To solve this, we need to recall the fundamental relationship between a matrix, its adjoint, and its determinant.
Key Relationship: Matrix, Adjoint, and Determinant
For any square matrix \(A\) of order \(n\), the product of the matrix and its adjoint is given by the formula:
\[A(\text{adj } A) = (\text{adj } A)A = (\det A)I_n\]where \(\det A\) is the determinant of matrix \(A\), and \(I_n\) is the identity matrix of order \(n\). This relationship is crucial for analyzing the given statements.
Given that \(A\) is a non-singular matrix, its determinant \(\det A \neq 0\). Also, we are given \(B = \text{adj } A\). Substituting \(B\) into the relationship, we get:
\[AB = BA = (\det A)I\]Now let's examine each statement based on this result.
Analyzing Statement 1: AB = BA
Statement 1 says \(AB = BA\). From the fundamental relationship \(A(\text{adj } A) = (\text{adj } A)A\), and substituting \(B = \text{adj } A\), we directly get \(AB = BA\). This shows that the product of a matrix and its adjoint is commutative.
Therefore, Statement 1 is correct.
Analyzing Statement 2: AB is a scalar matrix
Statement 2 says \(AB\) is a scalar matrix. We found that \(AB = (\det A)I\). Let's understand what a scalar matrix is.
A scalar matrix is a diagonal matrix where all the diagonal entries are equal. The identity matrix \(I\) is a diagonal matrix with all diagonal entries equal to 1. When we multiply \(I\) by a scalar value (\(\det A\) in this case), we get a matrix where the diagonal entries are all equal to that scalar value (\(\det A\)), and off-diagonal entries are zero.
For example, if \(A\) is a 3x3 matrix, \(I\) is \(I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\). Then \((\det A)I_3 = \begin{pmatrix} \det A & 0 & 0 \\ 0 & \det A & 0 \\ 0 & 0 & \det A \end{pmatrix}\). This matrix fits the definition of a scalar matrix.
Since \(A\) is non-singular, \(\det A \neq 0\), so \((\det A)I\) is indeed a scalar matrix with diagonal elements equal to \(\det A\).
Therefore, Statement 2 is correct.
Analyzing Statement 3: AB can be a null matrix
Statement 3 says \(AB\) can be a null matrix. We know that \(AB = (\det A)I\). For \(AB\) to be a null matrix (a matrix with all entries being zero), we would need \((\det A)I = 0\).
Since \(I\) is the identity matrix, its diagonal entries are 1 (non-zero). The only way for \((\det A)I\) to be the zero matrix is if the scalar multiplier \(\det A\) is zero.
However, the question explicitly states that \(A\) is a non-singular matrix. By definition, a non-singular matrix has a non-zero determinant, i.e., \(\det A \neq 0\).
Since \(\det A \neq 0\), \((\det A)I\) cannot be the zero matrix (null matrix).
Therefore, Statement 3 is incorrect.
Summary of Statements
| Statement | Content | Correctness | Reasoning |
|---|---|---|---|
| 1 | \(AB = BA\) | Correct | \(A(\text{adj } A) = (\text{adj } A)A\) holds true. |
| 2 | \(AB\) is a scalar matrix | Correct | \(AB = (\det A)I\), which is a scalar matrix when \(\det A \neq 0\). |
| 3 | \(AB\) can be a null matrix | Incorrect | \(AB = (\det A)I\); requires \(\det A = 0\), but A is non-singular (\(\det A \neq 0\)). |
Based on our analysis, Statement 1 and Statement 2 are correct, while Statement 3 is incorrect.
The option that includes only statements 1 and 2 is the correct answer.
Revision Table: Non-singular Matrix and Adjoint Properties
| Concept | Definition/Property | Relevance to Question |
|---|---|---|
| Non-singular Matrix (A) | \(\det A \neq 0\). Inverse \(A^{-1}\) exists. | Key condition preventing AB from being the null matrix. |
| Adjoint Matrix (adj A or B) | Transpose of the cofactor matrix. | The matrix B given in the question. |
| Fundamental Relation | \(A(\text{adj } A) = (\text{adj } A)A = (\det A)I\) | Core formula used to prove Statements 1 and 2. |
| Scalar Matrix | Diagonal matrix with equal diagonal elements. Form: \(kI\). | The structure of \(AB = (\det A)I\). |
| Null Matrix | Matrix with all elements equal to zero. | Statement 3 evaluates if AB can be this matrix. |
Additional Information: Matrix Inverse
The relationship \(A(\text{adj } A) = (\text{adj } A)A = (\det A)I\) is also used to find the inverse of a non-singular matrix. Since \(\det A \neq 0\), we can divide by \(\det A\):
\[\frac{1}{\det A} A(\text{adj } A) = \frac{1}{\det A}(\det A)I\] \[A \left(\frac{1}{\det A} \text{adj } A\right) = I\]Similarly, \(\left(\frac{1}{\det A} \text{adj } A\right) A = I\). By the definition of matrix inverse, if \(AC = CA = I\), then \(C = A^{-1}\).
Thus, for a non-singular matrix \(A\), its inverse \(A^{-1}\) is given by:
\[A^{-1} = \frac{1}{\det A} \text{adj } A\]This further highlights the importance of the determinant being non-zero for the inverse (and thus the adjoint relationship \(A(\text{adj } A) = (\det A)I\) to imply a non-null result) to exist in this context.
Important Questions from Adjoint and Inverse of a Square Matrix
Let A be a matrix of order 3 × 3 and |A| = 4. If |2adj(3A)| = 2α 3β, then what is the value of (α + β)?
If A is a square matrix, then what is adj (A -1 ) – (adj A) -1 equal to?
The matrix \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 1&3&2\\ 1&{{\rm{x}} - 1}&1\\ 2&7&{{\rm{x}} - 3} \end{array}} \right]\)
Will have inverse for every real number x except forIf $A = \begin{bmatrix} 2 & 7 \\ 1 & 5 \end{bmatrix}$, then what is $A + 3A^{-1}$ equal to, where $A$ is a matrix of order 2?
Find the value of $|adj (2 \cdot adj A)|$ if matrix $A$ is of the order of $3$ and $|A| = 15$.
