Question

If \(\mathrm{A}\) is a non \(-\) singular square matrix such that \(\mathrm{A}^{2}-7 \mathrm{~A}+5 \mathrm{I}=0\), then \(\mathrm{A}^{-1}=\) ____. (1) \(7 \mathrm{~A}-\mathrm{I}\) (2) \(\frac{7}{5} \mathrm{I}-\frac{1}{5} \mathrm{~A}\) (3) \(\frac{7}{5} \mathrm{I}+\frac{1}{5} \mathrm{~A}\) (4) \(\frac{\mathrm{A}}{5}-\frac{7}{5}\)

Short answer

Question: Given a square matrix A that satisfies the equation A² - 7A + 5I = 0, where I is the identity matrix, find the inverse of matrix A. Answer: The inverse of matrix A is A^(-1) = (2) \(\frac{7}{5} \mathrm{I}-\frac{1}{5} \mathrm{~A}\).

Step-by-step solution

Write the given equation

We are given the equation: A² - 7A + 5I = 0.

Rearrange the given equation to solve for A^(-1)

We need to isolate A^(-1) in the given equation. First, let's make A inverse on one side of the equation by adding -5I to both sides: A² - 7A = -5I. Now, let's factor A. Notice that A is a common factor for both terms on the left-hand side. Factor A out of the equation: A(A - 7I) = -5I. Now, we can divide both sides of the equation by (-5)A to isolate A^(-1): A^{-1} = \frac{-1}{5}(A - 7I). So, A^(-1) is equal to -\frac{1}{5}(A - 7I), which can be simplified to: A^(-1) = \frac{1}{5}(7I - A). Therefore, the inverse of matrix A is A^{-1} = \frac{1}{5}(7I - A). The correct choice is (2) \(\frac{7}{5} \mathrm{I}-\frac{1}{5} \mathrm{~A}\).

Key concepts

Understanding Non-Singular Square Matrices

A non-singular square matrix is a square matrix that has an interesting property: it is invertible. This means that for every non-singular square matrix A, there exists another matrix, known as the inverse of A, denoted as A^-1, such that when A is multiplied by A^-1, the result is the identity matrix, I. In mathematical terms, this relationship can be expressed as AA^-1 = A^-1A = I. It's like having an 'undo' operation for matrices.

But why is this important? Well, in linear algebra, being able to find an inverse is crucial for solving matrix equations, much like dividing is essential for solving algebraic equations. Keep in mind, not all square matrices have inverses; matrices that do not have an inverse are referred to as 'singular'. The given exercise focuses on a non-singular square matrix where finding its inverse is the key to solving the problem.

Solving Matrix Equations

Working with matrix equations is like solving a puzzle; each piece must fit perfectly for the entire picture to make sense. When faced with such equations, our objective is to find a matrix that satisfies the given conditions. Let's connect this to the exercise where we have the equation A² - 7A + 5I = 0. Solving this equation involves operations very similar to those we perform on regular numbers, but with matrices.

First, we rearrange the equation to get terms involving the matrix A on one side, akin to getting all x terms on one side of an algebraic equation. Then, by elegantly factoring and manipulating the matrix terms, we find ourselves with an expression for A^-1, the matrix we're trying to find.

Remember!

In these operations, order matters—matrix multiplication is not commutative, which means that AB does not necessarily equal BA.

The IIT Foundation Series

The IIT Foundation series is a collection of educational materials designed to build a robust foundational understanding of various scientific and mathematical concepts for students aspiring to face competitive exams like the IIT-JEE. These resources are meticulously crafted to develop problem-solving skills and conceptual clarity from the ground up.

The problem highlighted is typical of those found in an IIT Foundation series textbook, where the goal is not only to find the right answer but also to understand the underlying principles that govern the solution. It's a great illustration of how concept-oriented education, such as that encouraged by the IIT Foundation series, not only enhances academic knowledge but also sharpens analytical thinking, paving the way for success in prestigious exams and future academic endeavors.