Question 16 of 58beginner🔧 ApplyShort Answer2 marks

Find adjoint of each of the matrices:

[1234]\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
Correct Answer

[4−2−31]\begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}

Exercise: EXERCISE 4.4 | Q: 1 | (Page 17)
For More Understanding

Explanation

The textbook context provides a specific 'Remark' for finding the adjoint of a 2x2 matrix. It states that for a matrix A = [[a11, a12], [a21, a22]], the adjoint is obtained by interchanging a11 and a22, and changing the signs of a12 and a21. Applying this shortcut method is efficient and directly yields the result.

Solution Steps

  1. Step 1: Identify the elements of the matrix A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}. Here, a11=1a_{11} = 1, a12=2a_{12} = 2, a21=3a_{21} = 3, and a22=4a_{22} = 4.

  2. Step 2: Apply the rule for a square matrix of order 2: adj A=[a22−a12−a21a11]adj \ A = \begin{bmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{bmatrix}. This involves interchanging the diagonal elements and changing the signs of the off-diagonal elements.

  3. Step 3: Substitute the values into the formula: adj A=[4−2−31]adj \ A = \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}.

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