Problem 10 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The matrix \(A^{2}+4 A-5 I\), where \(I\) is identity matrix and \(A=\left[\begin{array}{cc}1 & 2 \\ 4 & -3\end{array}\right]\), equals [Online April 9, 2013] (a) \(4\left[\begin{array}{ll}2 & 1 \\ 2 & 0\end{array}\right]\) (b) \(4\left[\begin{array}{cc}0 & -1 \\ 2 & 2\end{array}\right]\) (c) \(32\left[\begin{array}{ll}2 & 1 \\ 2 & 0\end{array}\right]\) (d) \(32\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right]\)

Step-by-Step Solution

Answer

The correct answer is (a) \(4\begin{bmatrix} 2 & 1 \\ 2 & 0 \end{bmatrix}\).

Step 1: Calculate A²

First, compute the square of matrix \( A \):\[ A = \begin{bmatrix} 1 & 2 \ 4 & -3 \end{bmatrix} \]To find \( A^2 \), perform the matrix multiplication \( A \times A \):\[A^2 = \begin{bmatrix} 1 & 2 \ 4 & -3 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \ 4 & -3 \end{bmatrix} = \begin{bmatrix} (1)(1) + (2)(4) & (1)(2) + (2)(-3) \ (4)(1) + (-3)(4) & (4)(2) + (-3)(-3) \end{bmatrix} = \begin{bmatrix} 9 & -4 \ -8 & 17 \end{bmatrix} \]

Step 2: Compute 4A

Multiply the matrix \(A\) by 4:\[4A = 4 \times \begin{bmatrix} 1 & 2 \ 4 & -3 \end{bmatrix} = \begin{bmatrix} 4 & 8 \ 16 & -12 \end{bmatrix}\]

Step 3: Compute 5I

Calculate \(5\) times the identity matrix \(I\) of the same order:\[I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]\[5I = 5 \times \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix}\]

Step 4: Calculate A² + 4A - 5I

Now substitute the calculated values into the expression:\[A^2 + 4A - 5I = \begin{bmatrix} 9 & -4 \ -8 & 17 \end{bmatrix} + \begin{bmatrix} 4 & 8 \ 16 & -12 \end{bmatrix} - \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix} \]Perform matrix addition and subtraction:\[= \begin{bmatrix} 9+4-5 & -4+8-0 \ -8+16-0 & 17-12-5 \end{bmatrix} \]\[= \begin{bmatrix} 8 & 4 \ 8 & 0 \end{bmatrix}\]

Step 5: Compare with Options

The final result from the previous step was \(\begin{bmatrix} 8 & 4 \ 8 & 0 \end{bmatrix}\).Now compare this result with the given options:- Option (a): \(4\begin{bmatrix} 2 & 1 \ 2 & 0 \end{bmatrix}\) evaluates to \(\begin{bmatrix} 8 & 4 \ 8 & 0 \end{bmatrix}\).- Other options do not match.Therefore, the correct answer is (a).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that combines two matrices to produce a new matrix. To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Each element in the resulting matrix is calculated by taking the dot product of the respective row from the first matrix and column from the second matrix.
This means for each element, you multiply corresponding entries and sum them up:

• For matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and matrix \(B = \begin{bmatrix} e & f \ g & h \end{bmatrix}\), the product \(A \times B\) results in:
• \[ \begin{bmatrix} ae + bg & af + bh \ ce + dg & cf + dh \end{bmatrix} \]

Matrix multiplication is not commutative, meaning \(A \times B\) is not necessarily equal to \(B \times A\). However, it is associative, meaning \((A \times B) \times C = A \times (B \times C)\). This property is particularly important when calculating powers of a matrix, such as \(A^2 = A \times A\), which was used in the exercise above.

Identity Matrix

The identity matrix is a special kind of matrix that acts as a multiplicative identity in matrix algebra. When a matrix is multiplied by an identity matrix, it remains unchanged. This property is similar to how multiplying a number by one leaves it unchanged.

The identity matrix, usually denoted as \(I\), is a square matrix with ones on the main diagonal (from top left to bottom right) and zeros elsewhere. For a 2x2 identity matrix, it looks like this:

• \[ I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]

In operations, the identity matrix is particularly significant because:

• For any matrix \(A\), multiplying \(A \times I\) or \(I \times A\) will yield \(A\) itself.
• This property was employed in the original exercise when calculating \(5I\), multiplying the identity matrix by a scalar to facilitate the subtraction from other matrices.

Understanding the identity matrix helps unveil the structure and behavior of more complex matrix operations.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vector spaces and linear transformations. It is the mathematical backbone of many areas including computer science, engineering, physics, and data analytics. Central to linear algebra are matrices and vectors, which are used to represent and solve systems of linear equations, among other things.

Some key concepts in linear algebra include:

• Vectors: Objects that have both magnitude and direction. They can be represented as lists of numbers, which correspond to coordinates.
• Matrices: Rectangular arrays of numbers that can represent transformations or systems of equations. Matrices are used to encapsulate multiple equations and simplify calculations.
• Determinants & Eigenvalues: Determinants give important information about matrices, such as invertibility, while eigenvalues are pivotal in analyzing linear transformations.

In the context of the exercise, we used matrices to perform algebraic operations such as multiplication and addition. Using linear algebra principles, we manipulated these matrices to solve the given problem, illustrating how powerful and versatile the tools of linear algebra can be in solving complex problems.