If A = [[1, 2], [3, 4]], what is the adjoint of A?

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Q: If A = [[1, 2], [3, 4]], what is the adjoint of A?

Solution: The adjoint of A is [[4, -2], [-3, 1]].

Steps: 6

Step 1: Identify the matrix A. Here, A = [[1, 2], [3, 4]].
Step 2: Calculate the determinant of A. The determinant is calculated as (1*4) - (2*3) = 4 - 6 = -2.
Step 3: Find the matrix of minors. For A, the minors are: M11 = 4, M12 = 3, M21 = 2, M22 = 1. So, the matrix of minors is [[4, 3], [2, 1]].
Step 4: Apply the cofactor matrix by changing the signs according to the checkerboard pattern. The cofactor matrix is [[4, -3], [-2, 1]].
Step 5: Transpose the cofactor matrix. The transpose of [[4, -3], [-2, 1]] is [[4, -2], [-3, 1]].
Step 6: The result from the transpose is the adjoint of A. Therefore, the adjoint of A is [[4, -2], [-3, 1]].