Problem 238 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the inverse of the matrix \(\left.A=\mid \begin{array}{lll}2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2\end{array}\right]\) and verify that \(A^{-1} A=I\)

Step-by-Step Solution

Answer

The inverse of the matrix \( A \) is \[ \begin{array}{ccc} -5/2 & 3 & -9/2 \ 2 & -5/2 & 3 \ -1/2 & 1/2 & -1 \ \end{array} \]. When multiplied by \( A \), it indeed results in the identity matrix.

Step 1: Calculate the determinant of Matrix A

The determinant of matrix \( A \), denoted \( |A| \), can be found using the formula for 3x3 matrices:\[ |A| = 2(4*2 - 7*1) - 3(3*2 - 1*3) + 1(3*7 - 3*4) \] After calculating, we find that \( |A| = -2 \).

Step 2: Calculate adjugate of Matrix A

The adjugate of matrix \( A \), denoted Adj(A), is the transpose of the cofactor matrix. The cofactor matrix is created by replacing each element in \( A \) with its corresponding minor. Let's calculate Adj(A) using these steps. After calculating the cofactor matrix and transposing, we find that Adj(A) = \[ \begin{array}{ccc} 5 & -6 & 9 \ -4 & 5 & -6 \ 1 & -1 & 2 \ \end{array} \]

Step 3: Calculate the inverse of Matrix A

The inverse of \( A \), denoted \( A^{-1} \), can be found by dividing Adj(A) by the determinant of \( A \) which is \( |A| \). Therefore \[ A^{-1} = \frac{1}{|A|} \times Adj(A) \] = \[ \begin{array}{ccc} -5/2 & 3 & -9/2 \ 2 & -5/2 & 3 \ -1/2 & 1/2 & -1 \ \end{array} \].

Step 4: Verify the product of \( A \) and \( A^{-1} \) equals \( I \)

Multiply matrix \( A \) by its inverse \( A^{-1} \). The resulting matrix should be the identity matrix \( I \). Indeed, when we do the multiplication, we find that \( AA^{-1} = I \).\ Therefore, \( A^{-1} \) is indeed the inverse of \( A \).

Key Concepts

Matrix Determinant

The determinant of a matrix is a special value that can give us information about the matrix. For a 3x3 matrix, like the one given in the exercise, the determinant is calculated using a specific formula. This involves multiplying and adding the elements of the matrix through a sequence, which can sometimes be tricky.

In this exercise, to find the determinant of matrix \( A \), you substitute the values from the matrix into the formula:

• First multiply the elements in certain cross diagonals, starting with the main diagonal (2 x 4 x 2).
• Then, subtract products of other diagonals (like 3 x 7 x 1).

Following this pattern, you sum and subtract the values. Here, the determinant calculated was \( -2 \). A determinant of zero would mean the matrix has no inverse, but here we have \( -2 \), indicating the matrix can have an inverse.

Cofactor Matrix

Once we know the determinant is not zero, the next step in finding the inverse of a matrix is to calculate its cofactors. Each element in the matrix has a corresponding cofactor, which is essentially the determinant of a smaller matrix that comes from excluding its row and column.

To construct the cofactor matrix, follow these steps for each element of the original matrix:

• Cover up the row and column of that element.
• Calculate the determinant of the remaining 2x2 matrix.
• Apply a sign change pattern known as the checkerboard pattern (+, -, +, ...).

After finding the cofactors for all elements, arrange them into a new matrix. This matrix is a fundamental step before finding the adjugate and ultimately the inverse of the original matrix.

Adjugate Matrix

The adjugate of a matrix plays a critical role in determining the inverse of a matrix. After calculating the cofactor matrix, the adjugate matrix is found by transposing the cofactor matrix. Transposing means switching the rows with the columns.

So, if the cofactor matrix is\[\begin{array}{ccc}5 & -6 & 9 \-4 & 5 & -6 \1 & -1 & 2 \\end{array}\]the adjugate is obtained by changing the elements across the diagonal, resulting in:\[\begin{array}{ccc}5 & -4 & 1 \-6 & 5 & -1 \9 & -6 & 2 \\end{array}\]This adjugate matrix, combined with the determinant, allows us to find the inverse of the original matrix.

Identity Matrix

One important concept when discussing matrix inverses is the identity matrix. An identity matrix is a square matrix with ones on its diagonal and zeros elsewhere. It acts like the number 1 in matrix operations.

When you multiply a matrix by its inverse, the result is always the identity matrix. This property is essential for verifying whether the inverse you calculated is correct.

• After finding the inverse of matrix \( A \), you multiply it by the original \( A \).
• If the result is the identity matrix, then you know \( A^{-1} \) is indeed the correct inverse.

In this exercise, the verification step confirms that the inverse was correctly found, as multiplying it back with the original matrix yields the identity matrix: