Adjoint Of A Matrix

Step 1: Recall the 2 $%5Ctimes$ 2 formula. For $A%3D%5Cbegin%7Bbmatrix%7Da%26b%20%5C%5C%20c%26d%5Cend%7Bbmatrix%7D$ , $%5Coperatorname%7Badj%7D(A)%3D%5Cbegin%7Bbmatrix%7Dd%26-b%20%5C%5C%20-c%26a%5Cend%7Bbmatrix%7D.$ This comes from taking cofactors and then transposing the cofactor matrix (for a $2%5Ctimes2$ , that reduces to swapping $a%5Cleftrightarrow%20d$ and negating the off–diagonals). \\ [4pt]

Step 2: Apply it to $A%3D%5Cbegin%7Bbmatrix%7D1%262%20%5C%5C%203%264%5Cend%7Bbmatrix%7D$ . Here $a%3D1%2C%5C%20b%3D2%2C%5C%20c%3D3%2C%5C%20d%3D4$ . Therefore $%5Coperatorname%7Badj%7D(A)%3D%5Cbegin%7Bbmatrix%7D4%26-2%20%5C%5C%20%5B2pt%5D-3%261%5Cend%7Bbmatrix%7D.$ This matches option (A).

To find the adjoint of a matrix, we first calculate the cofactor matrix and then transpose it. Given matrix

A%20%3D%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%202%20%26%203%20%5C%5C%205%20%26%204%20%5Cend%7Bmatrix%7D%20%5Cright%5D

, we will find the cofactor matrix by finding the minors and then the cofactors: - For the (1,1) element: The minor is the determinant of

%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%204%20%5Cend%7Bmatrix%7D%20%5Cright%5D

, which is 4. The cofactor is

%2B4

. - For the (1,2) element: The minor is the determinant of

%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%205%20%5Cend%7Bmatrix%7D%20%5Cright%5D

, which is 5. The cofactor is

-5

. - For the (2,1) element: The minor is the determinant of

%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%203%20%5Cend%7Bmatrix%7D%20%5Cright%5D

, which is 3. The cofactor is

-3

. - For the (2,2) element: The minor is the determinant of

%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%202%20%5Cend%7Bmatrix%7D%20%5Cright%5D

, which is 2. The cofactor is

%2B2

. Thus, the cofactor matrix is:

%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%204%20%26%20-5%20%5C%5C%20-3%20%26%202%20%5Cend%7Bmatrix%7D%20%5Cright%5D

Now, we take the transpose of the cofactor matrix to get the adjoint matrix, which remains the same in this case. Therefore, the adjoint is:

%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%204%20%26%20-3%20%5C%5C%20-5%20%26%202%20%5Cend%7Bmatrix%7D%20%5Cright%5D

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About Adjoint Of A Matrix - BIHAR-BOARD-XII

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Adjoint Of A Matrix

High-Yield Trend

Chapter Questions 2 MCQs

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About Adjoint Of A Matrix - BIHAR-BOARD-XII

Frequently Asked Questions

Why focus on Adjoint Of A Matrix PYQs?

How to best use this analysis?

Chapter Questions
2 MCQs