# A real 2 × 2 non-singular matrix A with repeated eigenvalue is given as$$A = \left[ {\begin{array}{*{20}{c}} x&{ - 3.0}\\ {3.0}&{4.0} \end{array}} \right]$$where x is a real positive number. The value of x (rounded off to one decimal place) is _______

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## Answer (Detailed Solution Below) 10

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CT 1: Ratio and Proportion
1784
10 Questions 16 Marks 30 Mins

## Detailed Solution

Concept:

Non Singular Matrix: It is a matrix whose determinant ≠ 0

1. If A is any square matrix of order n, we can form the matrix [A – λI], where I is the nth order unit matrix. The determinant of this matrix equated to zero i.e. |A – λI| = 0 is called the characteristic equation of A.

2. The roots of the characteristic equation are called Eigenvalues or latent roots or characteristic roots of matrix A.

3. Sum of Eigenvalues is equal to the trace of that matrix.

4. The product of Eigenvalues of a matrix A is equal to the determinant of that matrix A.

Calculation:

$$A = \left[ {\begin{array}{*{20}{c}} x&{ - 3.0}\\ {3.0}&{4.0} \end{array}} \right]$$

To find eigen-values:

|A – λI| = 0

$$\left| {\begin{array}{*{20}{c}} x-λ&{ - 3.0}\\ {3.0}&{4.0-λ} \end{array}} \right|=0$$

|(4 - λ)(x - λ) - (-3)(3)| = 0

λ2 - (4 + x)λ + (4x + 9) = 0

As the roots are repeated (λ1 = λ2)

So according to quadratic equations property:

b2 - 4ac = 0

(4 + x)2 - 4(4x + 9) = 0

16 + x2 + 8x - 16x - 36 =0

x2 - 8x - 20 = 0

x2 + 2x - 10x + 20 = 0

x(x + 2) - 10(x + 2) = 0

x = -2 and x = 10

As x = -2 not possible (given)

So x = 10
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