Problem 65 - A Complete Resource Book in Mathematics for JEE Main 2017

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

Option (B) is correct: \(A^2 = I\).

Step 1: Examine Matrix A

Define the matrix \(A\) as provided: \(A=\begin{pmatrix} 0 & 0 & -1 \ 0 & -1 & 0 \ -1 & 0 & 0 \end{pmatrix}\). Observe that it is a 3x3 matrix with certain non-zero elements.

Step 2: Check if A is a Zero Matrix (Option A)

A zero matrix has all its elements as zeros. Check the elements of \(A\). Since \(A\) contains non-zero elements like \(-1\), option (A) is incorrect.

Step 3: Check if A² = I (Option B)

Compute the square of \(A\): Calculate \(A^2 = A \cdot A = \begin{pmatrix} 0 & 0 & -1 \ 0 & -1 & 0 \ -1 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 0 & -1 \ 0 & -1 & 0 \ -1 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} = I\). Thus, \(A^2 = I\), which means option (B) is correct.

Step 4: Check if A^{-1} does not exist (Option C)

A matrix has an inverse if and only if it is non-singular, which means its determinant is non-zero. Since \(A^2 = I\), \(A\) is clearly invertible, so its inverse exists. Hence, option (C) is incorrect.

Step 5: Check if A = -I (Option D)

Compare \(A\) with \(-I\) where \(I\) is the identity matrix: \(-I = \begin{pmatrix} -1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & -1 \end{pmatrix}\). Clearly, \(A\ eq -I\), thus option (D) is incorrect.

Key Concepts

Inverse of a Matrix

The inverse of a matrix is a fundamental concept in linear algebra. For a square matrix, the inverse is another matrix that, when multiplied with the original, yields the identity matrix. In mathematical terms, if matrix \( A \) has an inverse, it is denoted as \( A^{-1} \), and satisfies the equation \( A \cdot A^{-1} = I \) and \( A^{-1} \cdot A = I \). The identity matrix, \( I \), is like the number 1 in regular multiplication; it's the neutral element.

• Existence: A matrix must be non-singular to have an inverse. This means its determinant should be non-zero.
• Finding the Inverse: Various methods, such as Gaussian elimination or using the adjoint, can compute inverses. For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the inverse is \( \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \) if \( ad-bc eq 0 \).

Here, since it's determined \( A^2 = I \), matrix \( A \) clearly has an inverse because \( A \cdot A = I \). This confirms that the inverse exists, as shown in step 4 of the solution.

Matrix Multiplication

Matrix multiplication is crucial and somewhat different from regular multiplication. When multiplying matrices, we combine rows from the first matrix with columns from the second.

• Dimensions: The number of columns in the first matrix must be equal to the number of rows in the second. If matrix \( A \) is \( m \times n \) and matrix \( B \) is \( n \times p \), then their product \( AB \) will be an \( m \times p \) matrix.
• Calculation: To find each element \( c_{ij} \) of the resulting matrix \( C = AB \), calculate the dot product of the \( i \)-th row of \( A \) with the \( j \)-th column of \( B \).

In the exercise, multiplying matrix \( A \) with itself results in the identity matrix, \( I \). This means \( A \cdot A = I \), reinforcing the property \( A^2 = I \), thereby ensuring the matrix \( A \) can represent itself as a sort of 'dimension resetter'.

Identity Matrix

The identity matrix is a special form of a matrix, acting as the multiplicative identity in matrix algebra. It's comparable to the number 1 in regular arithmetic, as multiplying any matrix by the identity matrix will yield the original matrix unchanged.

• Form: It is a square matrix with ones on its diagonal and zeros elsewhere. For a 3x3 identity matrix, it looks like \( I = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} \).
• Properties: The identity matrix, for any square matrix \( A \) of the same size, satisfies \( A \cdot I = A \) and \( I \cdot A = A \).

In this exercise, calculating \( A^2 \) results in the identity matrix, \( I \). This characteristic shows that \( A \) is an involutory matrix, meaning when multiplied by itself, it returns the identity matrix.