Question: If A = [[1, 2], [3, 4]], what is the adjoint of A?
Options:
[[4, -2], [-3, 1]]
[[1, 3], [2, 4]]
[[2, 1], [4, 3]]
[[0, 0], [0, 0]]
Correct Answer: [[4, -2], [-3, 1]]
Solution:
The adjoint of A is [[4, -2], [-3, 1]].
If A = [[1, 2], [3, 4]], what is the adjoint of A?
Practice Questions
Q1
If A = [[1, 2], [3, 4]], what is the adjoint of A?
[[4, -2], [-3, 1]]
[[1, 3], [2, 4]]
[[2, 1], [4, 3]]
[[0, 0], [0, 0]]
Questions & Step-by-Step Solutions
If A = [[1, 2], [3, 4]], what is the adjoint of A?
Correct Answer: [[4, -2], [-3, 1]]
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]].
Adjoint of a Matrix – The adjoint (or adjugate) of a matrix is the transpose of its cofactor matrix, which is used in finding the inverse of a matrix.
Cofactor Calculation – To find the adjoint, one must calculate the cofactors of each element in the matrix, which involves determining the determinant of the submatrices.
Matrix Transposition – The final step in finding the adjoint is transposing the cofactor matrix.
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