Problem 19 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

The cofactor of the elements of the second row in determinant \(\left|\begin{array}{ccc}1 & 2 & 3 \\ -4 & 3 & 6 \\ 2 & -7 & 9\end{array}\right|\) is: (a) \(-39,3,11\) (b) \(-39,3,-11\) (c) \(39,3,11\) (d) \(39,-3,11\)

Step-by-Step Solution

Answer

The cofactors are \(-39, 3, 11\), answer (b).

Step 1: Understand the Determinant Layout

The determinant given is a 3x3 matrix: \[\left|\begin{array}{ccc}1 & 2 & 3 \ -4 & 3 & 6 \ 2 & -7 & 9\end{array}\right|.\] We are asked to find the cofactors of each element in the second row, which is \(-4, 3, 6\).

Step 2: Find the Minor of -4

The minor of an element is the determinant formed by deleting the row and column of that element. For -4 (in the second row and first column), delete the first column and second row: \[\left|\begin{array}{cc}2 & 3 \ -7 & 9\end{array}\right|.\] The determinant of this 2x2 matrix is calculated as \(2 \times 9 - 3 \times (-7) = 18 + 21 = 39\).

Step 3: Find the Cofactor of -4

The cofactor of an element is given by \((-1)^{i+j}\) times the minor of the element, where \(i\) and \(j\) refer to the row and column indices of the element. For \(-4\) at position \((2,1)\), the sign \((-1)^{2+1} = -1\). Thus, the cofactor of \(-4\) is \(-1 \times 39 = -39\).

Step 4: Find the Minor of 3

For 3 (second row, second column), delete the second row and column: \[\left|\begin{array}{cc}1 & 3 \ 2 & 9\end{array}\right|.\] Calculate the determinant as \(1 \times 9 - 3 \times 2 = 9 - 6 = 3\).

Step 5: Find the Cofactor of 3

For 3 at position \((2,2)\), use the sign \((-1)^{2+2} = 1\). Thus, the cofactor of 3 is \(1 \times 3 = 3\).

Step 6: Find the Minor of 6

For 6 (second row, third column), delete the third column and second row: \[\left|\begin{array}{cc}1 & 2 \ 2 & -7\end{array}\right|.\] Calculate the determinant as \(1 \times (-7) - 2 \times 2 = -7 - 4 = -11\).

Step 7: Find the Cofactor of 6

For 6 at position \((2,3)\), use the sign \((-1)^{2+3} = -1\). Thus, the cofactor of 6 is \(-1 \times -11 = 11\).

Key Concepts

3x3 Determinant

Understanding the concept of a 3x3 determinant can initially seem challenging, yet it is quite manageable with an organized approach. A 3x3 determinant is essentially a square matrix with three rows and three columns.
Each element in this matrix is used to perform calculations in a structured way to arrive at a specific numerical value known as the determinant. This value is crucial in various areas of linear algebra and can determine whether a matrix is invertible.

• The general form of a 3x3 matrix looks like this: \[\left|\begin{array}{ccc}a & b & c \d & e & f \g & h & i \end{array}\right|.\]

This discipline in structure allows us to systematically perform calculations for different mathematical operations like finding cofactors or solving systems of equations.

Matrix Minors

Matrix minors play a fundamental role in the calculation of determinants and cofactors. The minor of a particular element in a matrix is found by deleting the row and column that intersect at that element. This omission creates a smaller matrix, usually a 2x2 matrix for a 3x3 determinant.
For example, consider the element -4 in the second row and first column of a matrix.
Removing its row and column, we derive the minor matrix: \[\left|\begin{array}{cc}2 & 3 \-7 & 9\end{array}\right|.\]

• To obtain the minor, calculate the determinant of this reduced matrix, which is simplified as multiplying diagonals and subtracting the products: \[2 \times 9 - 3 \times (-7) = 18 + 21 = 39.\]

This minor will then be used in calculating the cofactor of the element.

Determinant Calculation

Determinant calculation is key when working with matrices, particularly for solving equations and understanding matrix properties. For a 3x3 matrix, the determinant is calculated using a specific rule involving elements and their corresponding cofactors.
In essence, the determinant of a 3x3 matrix can be found by choosing a row or column, and then summing the products of each element with its cofactor. For instance, using elements from the first row, the determinant is:
\[ a(ei - fh) - b(di - fg) + c(dh - eg). \]

• The step-by-step approach includes:

• Selecting an element and calculating its minor.
• Determining its cofactor by considering the sign \((-1)^{i+j}\).
• Repeating for each element of a chosen row or column and summing the terms.

Mastering these calculations helps build a strong foundation for more complex topics in linear algebra.