Problem 223 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the adjoint of the matrix \(A=\left[\begin{array}{ccc}1 & 0 & -1 \\ 3 & 4 & 5 \\ 0 & -6 & -7\end{array}\right]\) and verify that i. \(\quad A(a d j A)=|A| I_{3}=(a d j A) A\) ii. \(\quad|a d j A|=|A|^{2}\) iii. \(a d j A^{T}=(a d j A)^{T}\) iv. \(\operatorname{adj}(\operatorname{adj} A)=|A| A\).

Step-by-Step Solution

Answer

Firstly, iterate the given matrix A and calculate the cofactor of each element to obtain 'cofactor matrix'. Then, transpose this 'cofactor matrix' to get the adjoint of A. Verify the properties by following the mathematics rules given. Use the operations like determinant, transpose, and matrix multiplication. If all properties fulfill the given conditions, it means the calculation of the adjoint of matrix A is correct, and all properties are verified successfully.

Step 1: Find the Adjoint of Matrix A

To find the adjoint of matrix A, the cofactor matrix needs to be calculated first. Calculate the cofactor for each element and put it in the respective position. Once the cofactor matrix is obtained, take its transpose to find the adjoint. \[ adjA = CofactorMatrix(A)^T \]

Step 2: Verify Property i.

Multiply matrix \(A\) with adjoint \(A\) and verify if it results in a matrix of \(|A|I_{3}\) where \(|A|\) is the determinant of the matrix. It should also be proven that this equation is commutative, i.e., \(A*adjA = adjA*A\)

Step 3: Verify Property ii.

Calculate the determinant of adjoint of the matrix \(A\) and verify if it is equal to the square of the determinant of the original matrix. \[ |adjA|=|A|^2 \]

Step 4: Verify Property iii.

Calculate the transpose of the adjoint of matrix \(A\) and verify if it results in the transpose of the adjoint matrix.

Step 5: Verify Property iv.

Calculate the adjoint of the adjoint of matrix \(A\) and verify if it results in a matrix where each element is the product of the determinant of the original matrix and each element of the original matrix. \[ |A|A = adj( adjA) \]

Key Concepts

Cofactor Matrix

The foundation for finding the adjoint of a matrix is the cofactor matrix. To compute the cofactor of an element in a matrix, you first remove the row and column of that element to form a smaller matrix, known as the minor. You then take the determinant of this minor and apply a sign based on the position of the element in the matrix. The sign alternates in a checkerboard pattern, beginning with positive in the top-left element.

Here's how it's done:

• Determine the minor, which is the determinant of the smaller matrix formed.
• Apply the sign depending on the position (e.g., top-left is positive).
• The cofactor is this signed determinant.

Each element's cofactor is placed in the same corresponding position in the cofactor matrix. Once all cofactors are computed, they are arranged into a matrix, which we refer to as the cofactor matrix. This cofactor matrix is essential as it's the first step in eventually computing a matrix's adjoint.

Determinant

The determinant is a unique number that provides important information about a square matrix. It is used in various calculations, such as finding the inverse or determining if a matrix is invertible. The determinant of a 2x2 matrix is calculated as \( |A| = ad - bc\)whereas, for a 3x3 matrix, the process is more involved.To find the determinant of a 3x3 matrix, you expand along a row or column using cofactors from the first step:

• Choose any row or column to expand along; it’s common to begin with the first row for simplicity.
• Multiply each element of the row by its corresponding cofactor.
• Add the results, taking care of signs from the cofactor matrix.

This results in a single number, the determinant, which plays a vital role in matrix operations.

Transpose

The transpose of a matrix, denoted as \(A^T\), is achieved by swapping the matrix's rows and columns. This means that the element at the row \(i\) and column \(j\) in the original matrix becomes the element at the row \(j\) and column \(i\) in the transposed matrix.

The process for transposing a matrix is simple:

• For each row in the original matrix, create columns in the new matrix.
• Switchly smoothly between rows and columns to form new alignments.

The transpose is not just a mathematical trick; it preserves certain properties in computations and is essential in determining the adjoint, where the cofactor matrix needs to be transposed for the final step.

Matrix Multiplication

Matrix multiplication is a foundational operation in linear algebra that involves multiplying two matrices to result in a new matrix. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second.The process:

• Take each row element of the first matrix and multiply it with corresponding column elements of the second matrix.
• Add those products to get each element of the resulting matrix.
• This computation is repeated for every row and column combination.

It's noteworthy that matrix multiplication is not commutative, meaning \(A \times B eq B \times A\) in most cases. However, in verifying properties of the adjoint and original matrix, you may find situations where this multiplication exhibits symmetry, such as when proving \( A \times adj(A) = adj(A) \times A = |A|I_3 \). This property is particular to matrices and their adjoints.