Problem 71 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(A=\left[\begin{array}{ccc}5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\\ 0 & 0 & 5\end{array}\right]\), If \(\left|A^{2}\right|=25\) then \(|\alpha|\) equals (A) 5 (B) 1 (C) \(1 / 5\) (D) 5

Step-by-Step Solution

Answer

|\alpha| = \frac{1}{5}.

Step 1: Understand the problem

We need to find the value of \(|\alpha|\) given that \(|A^2| = 25\). This involves understanding determinants and matrix multiplication.

Step 2: Calculate the determinant of A

The matrix \(A\) is an upper triangular matrix, so its determinant is the product of its diagonal elements. Thus, \(|A| = 5 \cdot \alpha \cdot 5 = 25\alpha\).

Step 3: Find the determinant of A squared

Since \(|A^2| = (|A|)^2\), we have \(|A^2| = (25\alpha)^2 = 625\alpha^2\). We set this equal to 25: \ 625\alpha^2 = 25.

Step 4: Solve the equation for alpha

Solve \(625\alpha^2 = 25\) for \(\alpha^2\). First, divide both sides by 625: \ \alpha^2 = \frac{25}{625} = \frac{1}{25} \ Taking square roots of both sides gives \(|\alpha| = \frac{1}{5}\).

Step 5: Conclude the value of alpha

The value of \(|\alpha|\) that satisfies the condition \(|A^2| = 25\) is \(\frac{1}{5}\). So, the answer is option (C).

Key Concepts

Matrix Multiplication

Matrix multiplication is a systematic way of combining two matrices to produce another matrix. This operation isn’t as simple as multiplying individual numbers. Instead, it involves the rows of the first matrix and the columns of the second matrix. To effectively multiply two matrices:

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• The elements in each row of the first matrix are multiplied by the corresponding elements in each column of the second matrix, and their products are summed to form each element of the resulting matrix.
• This process continues for each row of the first matrix.

The matrices don’t have to be square, and matrix multiplication is not commutative, meaning that \(A \times B eq B \times A\). Understanding the intricacies of matrix multiplication is essential in linear algebra and applications such as physics, computer graphics, and statistics.

Upper Triangular Matrix

An upper triangular matrix is a special type of square matrix. In this matrix, all elements below the main diagonal are zero. This characteristic simplifies many calculations, such as finding determinants.

• The main diagonal of an upper triangular matrix consists of the first elements of each row starting from the top left.
• The determinant of an upper triangular matrix is simply the product of its diagonal elements. This makes calculating the determinant much easier compared to a general matrix.

For example, the matrix in the exercise is upper triangular because all the elements beneath the diagonal are zeros. Recognizing this pattern allows for quicker computations in both theoretical problems and practical applications.

Square Root Equation

Square root equations often appear when solving for unknown variables that are squared in algebraic expressions. The equation generally takes the form \(x^2 = a\), where the solution involves solving for \(x\) by taking the square root.

• The principal square root is the non-negative solution, but every positive number actually has two square roots: a positive and a negative one.
• When dealing with equations such as \(x^2 = 1\), the solutions are \(x = 1\) and \(x = -1\).
• Geometrically, the square root of \(x^2 = a\) is the side length of a square whose area is \(a\).

In the provided solution, taking square roots simplifies the problem and indicates that \(|\alpha| = \frac{1}{5}\). Solving square root equations is a fundamental skill in algebra, needed to simplify more complex problems efficiently.