Problem 251 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Prove that the matrix \(A=\left|\begin{array}{ccc}2 & -2 & -4 \\ -1 & 3 & 4 \\\ 1 & -2 & -3\end{array}\right|\) is idempotent.

Step-by-Step Solution

Answer

The matrix \(A\) is idempotent because \(A^2 = A\), i.e., when the matrix \(A\) is squared, the result is the matrix \(A\) itself.

Step 1: Write down the given

Given matrix \(A = \begin{bmatrix}2 & -2 & -4 \ -1 & 3 & 4\ 1 & -2 & -3\end{bmatrix}\)

Step 2: Compute the product \(AA\)

The product of \(A\) and \(A\) is computed as follows: \(AA = \begin{bmatrix}2 & -2 & -4 \ -1 & 3 & 4\ 1 & -2 & -3\end{bmatrix} \cdot \begin{bmatrix}2 & -2 & -4 \ -1 & 3 & 4\ 1 & -2 & -3\end{bmatrix} = \begin{bmatrix}2 & -2 & -4 \ -1 & 3 & 4\ 1 & -2 & -3\end{bmatrix}\)

Step 3: Check if \(A = AA\)

Comparing the original given matrix \(A\) with the computed product \(AA\), we can see that they are identical. Thus, \(A\) is indeed an idempotent matrix.

Key Concepts

Matrix Multiplication

Matrix multiplication is a key operation in linear algebra. It involves taking two matrices and producing another matrix by following a specific process. The number of columns in the first matrix must match the number of rows in the second matrix for multiplication to be possible. This is a fundamental requirement and is crucial for error-free computations.
To multiply matrices, align the rows of the first matrix with the columns of the second matrix. Then, calculate the sum of the products of corresponding elements. This sum gives you an element in the resulting matrix. For example, consider matrices \(A\) and \(B\). The element at the first row and first column of the resulting matrix is obtained by multiplying elements from the first row of \(A\) with corresponding elements in the first column of \(B\), then summing them up.

• Matrix multiplication is not commutative. This means \(AB eq BA\) in general.
• It is associative, which implies \((AB)C = A(BC)\).
• Matrix multiplication is distributive over addition: \(A(B + C) = AB + AC\).

Linear Algebra

Linear Algebra is a branch of mathematics concerned with vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, but also extends to abstract algebraic settings.
Understanding matrices is vital in linear algebra as they represent linear transformations. Given a matrix, it can transform a vector by multiplying them, thereby mapping it to a new vector space. This is particularly useful in solving systems of linear equations.
Application is broad, spanning across sciences and engineering, as linear algebra forms the backbone for machine learning algorithms, facial recognition technologies, and even 3D graphics.

• Vectors can be represented as matrices of one column.
• Operations include addition, subtraction, and scalar multiplication.
• More complex operations involve determinants, eigenvalues, and eigenvectors.

Understanding and mastering the fundamentals of linear algebra is a game-changer for advanced mathematics study.

Properties of Matrices

Matrices have specific properties that help classify and understand their behavior under operations like multiplication. An important property is idempotency, particularly relevant to this exercise.

An idempotent matrix is a matrix that, when multiplied by itself, yields the same matrix again. In mathematical terms, a matrix \(A\) is idempotent if \(A^2 = A\). This means, after performing the matrix multiplication \(A \times A\), the result should mirror matrix \(A\).

• Idempotent matrices form an interesting study since they represent operations remaining unchanged when applied repeatedly.
• Such matrices often find applications in statistics, especially in regression analysis.
• Other properties of matrices include symmetry, orthogonality, and invertibility, each with their own conditions and implications in matrix theory.

Understanding these properties helps us identify how matrices behave under different operations, which is foundational in both theoretical and applied linear algebra.