Problem 77 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Without expanding at any stage show that \(\left|\begin{array}{ccc}x^{2}+x & x+1 & x-2 \\ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{array}\right|=x A+B\), where \(A\) and \(B\) are determinants of order 3 not involving \(x\).

Step-by-Step Solution

Answer

Upon following the above steps, the matrix will be reduced to the form \(x \cdot A + B\), without fully expanding it. Matrix A and B can be identified in the final form.

Step 1 - Multiply rows by scalar

The prime step will involve scalar multiplication to induce zeros in the matrix for simplification. Multiply the first row by -2 and add it to the second row.

Step 2 - Produce Zeros

Next, keep the first and third row same and replace the second row with the result from step 1. This will create zeros in the second row, which will simplify the process.

Step 3 - Simplify matrix

Apply row operations to simplify the matrix i.e., subtract third row from the first row.

Step 4 - Factor Out

Now, the matrix is broken down into simpler form. Finally, factor an \(x\) out of the first row.

Step 5 - Identify the Matrices A and B

The final matrix will now resemble the form \(x \cdot A + B\), where A and B are simpler 3x3 matrices. Identify the resulting A and B.

Key Concepts

Matrix Operations

Matrix operations involve several processes that allow us to manipulate matrices to solve complex mathematical problems. In this exercise, the goal is to find a representation of a determinant using matrix operations. Here are some key operations we use:

• Addition: Summing up corresponding elements of two matrices.


• Scalar multiplication: Multiplying each element of a matrix by a constant.


• Row operation: Changing the rows of a matrix through operations like swapping, adding, and scalar multiplication.

These operations help transform matrices into forms that are more suitable for calculation, such as simpler matrices to determine properties like the determinant.

3x3 Matrices

A 3x3 matrix has three rows and three columns. Understanding these matrices is crucial as they often appear in mathematical problems that involve systems of equations. The matrix from the exercise looks like this: \[\begin{bmatrix}x^2 + x & x+1 & x-2 \2x^2 + 3x - 1 & 3x & 3x-3 \x^2 + 2x + 3 & 2x-1 & 2x-1\end{bmatrix}\]To simplify calculating the determinant, the matrix can be split into separate components (e.g., factoring out common variables like \(x\)) and through other operations like row replacements. The structure of a 3x3 matrix gives us a clear framework to apply all the necessary operations.

Scalar Multiplication

Scalar multiplication in matrices involves multiplying every element within the matrix by a single number, known as a scalar. In the provided exercise, the first step used scalar multiplication to assist in simplifying the matrix.Suppose you have a matrix \(M\) and a scalar \(c\), the product \(cM\) means each entry of \(M\) is multiplied by \(c\). For example, multiplying the first row by \(-2\) helped generate zeros when combined with other rows, allowing easier computation of the resulting determinant.

Row Operations

Row operations are transformations that simplify matrix computations, particularly in solving systems and finding determinants. They include three primary types: swapping rows, multiplying a row by a nonzero scalar, and adding or subtracting multiples of one row to another.In the exercise, several row operations were used:

• Multiply row 1 by \(-2\) and add it to row 2.


• Subtract the third row from the first row.

These operations allow us to alter the matrix without changing its determinant, which can simplify calculations significantly. They are integral to tasks like Gaussian elimination and other techniques used in linear algebra.