Question 21 of 58beginner🔧 ApplyShort Answer2 marks

Find the inverse of each of the matrices (if it exists):

[−15−32]\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}
Correct Answer

Let A = [−15−32]\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}

Step 1: Find |A| |A| = (-1)(2) - (5)(-3) = -2 + 15 = 13

Since |A| = 13 ≠\neq 0, the inverse exists.

Step 2: Find cofactors A₁₁ = 2, A₁₂ = 3, A₂₁ = -5, A₂₂ = -1

Step 3: Find adj A adj A = [2−53−1]\begin{bmatrix} 2 & -5 \\ 3 & -1 \end{bmatrix}

Step 4: Find A⁻¹ A⁻¹ = 1∣A∣\frac{1}{|A|} × adj A = 113[2−53−1]\frac{1}{13} \begin{bmatrix} 2 & -5 \\ 3 & -1 \end{bmatrix}

Exercise: EXERCISE 4.4 | Q: 6 | (Page 17)
For More Understanding

Explanation

The question asks to find the inverse of a 2×22 \times 2 matrix. From the textbook context, we use the formula A⁻¹ = (1/|A|) × adj A, where adj A is the transpose of the cofactor matrix. For a 2×22 \times 2 matrix, the cofactors are found by taking the determinant of the remaining element when that row and column are removed, with appropriate sign changes. The determinant is non-zero (13), so the inverse exists.

Solution Steps

  1. Step 1: Calculate determinant |A| = (-1)(2) - (5)(-3) = 13

  2. Step 2: Since |A| ≠\neq 0, inverse exists

  3. Step 3: Find cofactors: A₁₁ = 2, A₁₂ = 3, A₂₁ = -5, A₂₂ = -1

  4. Step 4: Form adj A = transpose of cofactor matrix = [[2, -5], [3, -1]]

  5. Step 5: Apply formula A⁻¹ = (1/13) × adj A

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