Problem 65 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(A=\left(\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right)\). The only correct statement about the matrix \(A\) is (A) \(A\) is a zero matrix (B) \(A^{2}=I\) (C) \(A^{-1}\) does not exist (D) \(A=(-1) I\), where \(I\) is a unit matrix

Step-by-Step Solution

Answer

(B) \(A^2 = I\).

Step 1: Identify the Elements of Matrix A

Matrix \(A\) is given as:\[A = \begin{pmatrix} 0 & 0 & -1 \ 0 & -1 & 0 \ -1 & 0 & 0 \end{pmatrix}\]It is a 3x3 matrix with specific entries.

Step 2: Evaluate Each Statement

We need to check each provided statement (A, B, C, D) one by one to determine which one is true.

Step 3: Check if A is a Zero Matrix (Option A)

A zero matrix has all its elements as zero. Clearly, matrix \(A\) is not a zero matrix because it contains non-zero elements like \(-1\). Thus, Option A is false.

Step 4: Check if A^2 = I (Option B)

Calculate \(A^2\) and check if it equals the identity matrix \(I\):\[A^2 = \begin{pmatrix} 0 & 0 & -1 \ 0 & -1 & 0 \ -1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & -1 \ 0 & -1 & 0 \ -1 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} = I\]Since \(A^2 = I\), Option B is correct.

Step 5: Check if A^-1 Does Not Exist (Option C)

Since \(A^2 = I\), it implies \(A\) is invertible and its inverse is \(A\) itself. Therefore, \(A^{-1}\) exists, making Option C false.

Step 6: Check if A = (-1)I (Option D)

Matrix \((-1)I\) should have \(-1\) along the diagonal and zeros elsewhere:\[(-1)I = \begin{pmatrix} -1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & -1 \end{pmatrix}\]Since \(A\) looks different than \((-1)I\), with distinct non-diagonal elements, Option D is false.

Key Concepts

Inverse of a Matrix

The inverse of a matrix is an important concept in linear algebra. For a square matrix to have an inverse, it must be invertible or non-singular.
In simple terms, if there exists another matrix that, when multiplied with the original matrix, results in the identity matrix, then that original matrix has an inverse.
Here's how it works:

• For matrix \(A\), there is an inverse \(A^{-1}\) such that \(A \cdot A^{-1} = I\) and \(A^{-1} \cdot A = I\).
• The identity matrix \(I\) is the equivalent of number 1 in matrices, meaning it does not change the matrix it's multiplied by.
• If no such inverse exists, then the matrix is singular.

In the given exercise, since \(A^2 = I\), matrix \(A\) has an inverse, and the inverse is \(A\) itself. Therefore, statement (C) is incorrect.

Identity Matrix

An identity matrix is a special type of square matrix that acts as the multiplicative identity for matrices, meaning any matrix multiplied by it remains unchanged.
The identity matrix \(I\) for a 3x3 matrix looks like this:

• It's a diagonal matrix with ones on its main diagonal and zeros elsewhere.
• Mathematically represented as \(I = \begin{pmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{pmatrix}\).

In the exercise, it is shown that multiplying \(A\) with itself (i.e., calculating \(A^2\)) results in the identity matrix \(I\). This validates statement (B), demonstrating how powerful a concept the identity matrix is in checking matrix properties.

Zero Matrix

A zero matrix is a matrix where all elements are zero.
It serves as the additive identity in matrix analysis, which means when you add any matrix to a zero matrix, the original matrix is unchanged.
Key features of a zero matrix:

• All entries are zero, and its size can vary.
• A zero matrix is often denoted by \(O\).

Considering the definition, the matrix \(A\) from the exercise cannot be a zero matrix since it contains non-zero elements.
Thus, statement (A) is false. Understanding zero matrices helps emphasize the unique properties of different types of matrices in linear algebra.