Problem 76 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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) \(A^{2}=I\) (b) \(A=(-1) I\), where \(I\) is a unit matrix (c) \(A^{-1}\) does not exist (d) \(A\) is a zero matrix

Step-by-Step Solution

Answer

(a) \(A^2=I\) is true; others are false.

Step 1: Understand the properties of matrix A

Matrix \( A \) is given as \( \begin{pmatrix} 0 & 0 & -1 \ 0 & -1 & 0 \ -1 & 0 & 0 \end{pmatrix} \). It does not appear to be a zero matrix because it contains non-zero elements.

Step 2: Evaluate \(A^2\)

Calculate \(A^2 = A \cdot A\): \[ A^2 = \begin{pmatrix} 0 & 0 & -1 \ 0 & -1 & 0 \ -1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & -1 \ 0 & -1 & 0 \ -1 & 0 & 0 \end{pmatrix} \]Calculate each element of the resulting matrix, ensuring the resulting product is the identity matrix.- First row, first column: \( 0 \cdot 0 + 0 \cdot 0 + (-1)\cdot(-1) = 1 \) - First row, second column: \( 0 \cdot 0 + 0 \cdot (-1) + (-1) \cdot 0 = 0 \) - First row, third column: \( 0 \cdot (-1) + 0 \cdot 0 + (-1) \cdot 0 = 0 \)- Repeat for all rows and columns to get: \[ A^2 = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} = I \]Thus, \( A^2 = I \), which is a true statement.

Step 3: Evaluate \(A = (-1)I\)

The matrix \( A \) is \( \begin{pmatrix} 0 & 0 & -1 \ 0 & -1 & 0 \ -1 & 0 & 0 \end{pmatrix} \), while \((-1)I\) would be \( \begin{pmatrix} -1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & -1 \end{pmatrix} \). Clearly, \( A \) is not equal to \((-1)I\).

Step 4: Check if \(A^{-1}\) exists

If \( A^{-1} \) exists, \( A \) should be invertible. We know \( A^2 = I \), meaning \( A \) is its own inverse. Therefore, the inverse of \( A \) does exist and is equal to \( A \) itself.

Step 5: Confirm \(A\) is not a zero matrix

Matrix \( A \) is \( \begin{pmatrix} 0 & 0 & -1 \ 0 & -1 & 0 \ -1 & 0 & 0 \end{pmatrix} \), which contains multiple non-zero elements. Hence, \( A \) is not a zero matrix.

Key Concepts

Identity Matrix

An identity matrix is a special kind of square matrix. It acts like the number one does in multiplication for real numbers. In mathematical terms, for a square matrix of size \( n \times n \), the identity matrix, denoted as \( I \), has a size of \( n \times n \) and contains ones on the diagonal from the top left to the bottom right corner. All other elements are zero.

• For example, a \( 3 \times 3 \) identity matrix looks like this: \[ I = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} \]
• Multiplying any matrix by the identity matrix leaves the original matrix unchanged, similar to multiplying a number by one.

This property is fundamental in matrix algebra and establishes a basis for understanding matrix multiplication and inverses.

Matrix Inverse

The inverse of a matrix \( A \) is another matrix \( A^{-1} \) such that when they are multiplied together, they produce the identity matrix: \( AA^{-1} = I \). This can be thought of similarly to how a reciprocal works for numbers; multiplying a number by its reciprocal results in one.

• Not all matrices have inverses; only square matrices that are non-singular or invertible do.
• For example, if \( A \) is \[ \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}, \] then \( A^{-1} \) would be a matrix such that \[ AA^{-1} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}. \]

Understanding matrix inverses is critical for solving equations involving matrices, just like finding the reciprocal of a number is necessary for division.

Matrix Multiplication

Matrix multiplication is different from standard number multiplication. It's essential to understand how to perform it to solve matrix equations accurately. The fundamental rule is that two matrices can be multiplied only if the number of columns in the first matrix matches the number of rows in the second.

• The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
• Each element is calculated as a dot product of the corresponding row from the first matrix and column from the second.

For example, if \( A \) is a \( 2 \times 3 \) matrix and \( B \) is a \( 3 \times 4 \) matrix, then the product \( AB \) will be a \( 2 \times 4 \) matrix.Matrix multiplication is not commutative, meaning \( AB \) might not be equal to \( BA \). This property differentiates it from regular multiplication and underlines the importance of order when dealing with matrices.

Matrix Properties

Matrices have numerous properties that make them unique and quite powerful in algebra. Whether dealing with identities or calculating inverses, knowing these properties can aid in solving matrix-related problems.

• Commutative Property: Unlike regular numbers, matrix multiplication does not generally follow the commutative law, i.e., \( AB eq BA \).
• Associative Property: That means for matrices \( A \), \( B \), and \( C \), the equation \( (AB)C = A(BC) \) holds.
• Distributive Property: Like numbers, matrices follow distributive laws over addition: \( A(B + C) = AB + AC \).

Understanding these properties helps simplify complex matrix operations and makes it easier to manage and solve algebraic equations involving matrices.