Problem 221 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the adjoint of \(\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3\end{array}\right]\).

Step-by-Step Solution

Answer

The adjoint of the given matrix is \[\begin{array}{ccc} -9 & -5 & 3 \ 5 & 4 & 2 \ 3 & 1 & 0 \end{array}\].

Step 1: Computations of cofactors for each element

Determine the cofactor for each element in the matrix. We'll use the formula \(-1^{i+j} \cdot M_{ij}\) to calculate cofactor of the element in the i-th row and j-th column. Here \(M_{ij}\) denotes the determinant of the matrix obtained by removing the i-th row and j-th column from the original matrix. After this step, we get the cofactor matrix: \[\begin{array}{ccc}-9 & 5 & 3 \ -5 & 4 & 1 \ 3 & 2 & 0 \end{array}\]

Step 2: Tranpose of the cofactor matrix

Find the transpose of the cofactor matrix obtained from the previous step. This is done by interchanging rows and columns of the cofactor matrix. This gives the adjoint of the original matrix:\[\begin{array}{ccc}-9 & -5 & 3 \ 5 & 4 & 2 \ 3 & 1 & 0 \end{array}\]

Step 3: Final Result

The matrix obtained in step 2 above is the adjoint of the given matrix. To confirm this, the original matrix and its adjoint can be multiplied. If the result is the determinant of the original matrix times the identity matrix, then the matrix from step 2 is indeed the adjoint.

Key Concepts

Matrix Cofactors

Matrix cofactors play an essential role in determining the adjoint of a matrix. A cofactor is a number that you get by removing one row and one column from a larger square matrix and calculating the determinant of the smaller matrix that's left behind. Here's a simplified way to understand it:

• Start by looking at an element in the matrix. Identify its position by its row (\(i\)) and column (\(j\)) numbers.
• Remove the row and column that contain this element. This leaves you with a smaller matrix called a minor.
• Calculate the determinant of this minor. This value is part of the cofactor.
• Multiply this determinant by \((-1)^{i+j}\). The use of \((-1)^{i+j}\) gives the cofactor its correct sign, depending on the position of the element in the matrix.

By calculating the cofactors for each element and arranging them in a new matrix, you get the cofactor matrix. This matrix is vital as it helps us find the adjoint of the original matrix.

Transpose of a Matrix

The transpose of a matrix is a straightforward yet crucial concept in matrix algebra. It involves swapping the rows and columns of a matrix. This means that the element which is in the \(i\)-th row and \(j\)-th column in the original matrix will be in the \(j\)-th row and \(i\)-th column in the transposed matrix. Here's how you can think about it:

• Take every row of the matrix and turn it into a column.
• The first row becomes the first column, the second row becomes the second column, and so on.

This simple operation is incredibly useful when calculating the adjoint of a matrix, as the adjoint is the transpose of the cofactor matrix. Transposing ensures the elements are correctly positioned to perform further operations, like matrix multiplication.

Matrix Determinant

A matrix determinant is a special number that can be calculated from a square matrix. It's like a fingerprint of a matrix, providing vital information about the matrix's properties such as whether it's invertible. Here's what you need to know:

• The determinant is computed using a structured formula and can involve recursive calculations for larger matrices.
• For a 2x2 matrix, the determinant is straightforward: \(det\begin{bmatrix}a & b \ c & d\end{bmatrix} = ad - bc\).
• For larger matrices, you often break them down into smaller 2x2 matrices by using methods such as expansion by minors. This involves cofactors calculated from smaller matrices.

The matrix's determinant is crucial when calculating the adjoint and other operations like finding the inverse. The adjoint is used to find the inverse of a matrix, and together with the determinant, they can tell us if a matrix is invertible. A non-zero determinant means a matrix has an inverse, while a zero determinant says it doesn't.