Understanding the Problem: Equating Determinants

The question asks us to find the possible values of the variable 'a' by equating the determinants of two different 2x2 matrices. We are given that the determinant of the matrix \(\left| {\begin{array}{*{20}{c}} 5&a\\ a&2 \end{array}} \right|\) is equal to the determinant of the matrix \(\left| {\begin{array}{*{20}{c}} 2&1\\ 3&2 \end{array}} \right|\). To solve this, we first need to calculate the determinant of each matrix and then set up an equation.

Calculating the Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix \(\left| {\begin{array}{*{20}{c}} p&q\\ r&s \end{array}} \right|\) is calculated using the formula:

\(\text{determinant} = (p \times s) - (q \times r)\)

This involves multiplying the elements on the main diagonal (top-left to bottom-right) and subtracting the product of the elements on the anti-diagonal (top-right to bottom-left).

Calculating the Determinants for the Given Matrices

Let's calculate the determinant for each matrix given in the problem.

Determinant of the Left-Hand Side Matrix

The left-hand side matrix is \(\left| {\begin{array}{*{20}{c}} 5&a\\ a&2 \end{array}} \right|\). Using the formula for a 2x2 determinant:

\(\text{Determinant}_1 = (5 \times 2) - (a \times a)\)

\(\text{Determinant}_1 = 10 - a^2\)

Determinant of the Right-Hand Side Matrix

The right-hand side matrix is \(\left| {\begin{array}{*{20}{c}} 2&1\\ 3&2 \end{array}} \right|\). Using the formula for a 2x2 determinant:

\(\text{Determinant}_2 = (2 \times 2) - (1 \times 3)\)

\(\text{Determinant}_2 = 4 - 3\)

\(\text{Determinant}_2 = 1\)

Forming and Solving the Equation to Find 'a' Values

We are given that the two determinants are equal. So, we can set up the equation:

\(\text{Determinant}_1 = \text{Determinant}_2\)

\(10 - a^2 = 1\)

Now, we need to solve this equation for 'a'. We can rearrange the equation to isolate \(a^2\):

\(10 - 1 = a^2\)

\(9 = a^2\)

Or, rewriting it with \(a^2\) on the left:

\(a^2 = 9\)

To find the values of 'a', we need to take the square root of both sides of the equation:

\(a = \pm \sqrt{9}\)

\(a = \pm 3\)

This means that 'a' can be either +3 or -3. These are the two possible values of 'a' that make the determinant of the first matrix equal to the determinant of the second matrix. We successfully used the determinant calculation to solve the equation and find the unknown values.

Summary of the Solution Process

Here is a quick summary of the steps taken to find the values of a:

• Understood the problem involves equating two determinants.
• Recalled the formula for calculating the determinant of a 2x2 matrix.
• Calculated the determinant of the first matrix, resulting in \(10 - a^2\).
• Calculated the determinant of the second matrix, resulting in \(1\).
• Set up the equation \(10 - a^2 = 1\).
• Solved the quadratic equation \(a^2 = 9\) for 'a'.
• Determined the possible values of 'a' are \(\pm 3\).

Calculating the determinant is a fundamental skill in matrix algebra, often used in solving systems of linear equations and finding eigenvalues. This problem demonstrates a simple application of the determinant concept.