Problem 52 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the matrices \(\mathrm{A}=\left[\begin{array}{ccc}1 & 1 & 2 \\ 1 & 3 & 4 \\\ 1 & -1 & 3\end{array}\right], \mathrm{B}=\operatorname{adj} \mathrm{A}\) and \(\mathrm{C}=3 \mathrm{~A}\), then \(\frac{|\operatorname{adj} \mathrm{B}|}{|\mathrm{C}|}\) is equal to: \([\) Jan. \(9, \mathbf{2 0 2 0}(\mathrm{I})]\) (a) 8 (b) 16 (c) 72 (d) 2

Step-by-Step Solution

Answer

The ratio \(\frac{|adj(B)|}{|C|}\) is equal to 8.

Step 1: Calculate Determinant of A

Given the matrix \[ A = \begin{bmatrix} 1 & 1 & 2 \ 1 & 3 & 4 \ 1 & -1 & 3 \end{bmatrix} \]we calculate its determinant using the formula for a 3x3 matrix:\[ |A| = a(ei-fh) - b(di-fg) + c(dh-eg) \]where the elements of A in their respective positions are: \\(a=1, b=1, c=2, d=1, e=3, f=4, g=1, h=-1, i=3 \).Calculate:\[ |A| = 1(3 \times 3 - 4 \times (-1)) - 1(1 \times 3 - 4 \times 1) + 2(1 \times (-1) - 3 \times 1) \]\[ = 1(9 + 4) - 1(3 - 4) + 2(-1 - 3) \]\[ = 13 - 1(-1) + 2(-4) \]\[ = 13 + 1 - 8 \]\[ = 6 \] So, \(|A| = 6\).

Step 2: Calculate Determinant of C

Using the property that the determinant of a scalar multiplied by a matrix is the scalar raised to the matrix dimension times the determinant of the matrix:\[ C = 3A \]Thus:\[ |C| = 3^3|A| = 27 \times 6 = 162 \]

Step 3: Properties of Adjugate and Determinant

The property of the adjugate of matrix \(B\) (which is \(adj(A)\)) is that \[ |adj(B)| = |B|^{(n-1)} = {(|A|^{(n-1)})}^{(n-1)} \]where \(n\) is the order of the matrix, here \(n = 3\).Thus,\[ |B| = |adj(A)| = |A|^{2} \]\[ |adj(B)| = |B|^{2} = (|A|^2)^2 = |A|^4 \]

Step 4: Calculate |adj(B)|

Substitute \(|A| = 6\) into the expression for \(|adj(B)|\):\[ |adj(B)| = |A|^4 = 6^4 = 1296 \]

Step 5: Calculate Ratio of Determinants

Calculate the ratio: \[ \frac{|adj(B)|}{|C|} = \frac{1296}{162} \] Simplifying:\[ \frac{1296}{162} = 8 \]

Step 6: Conclude the Answer

Thus, the answer to \(\frac{|adj(B)|}{|C|}\) is 8.

Key Concepts

Adjugate Matrix

Before diving into the depths of the adjugate matrix, let’s establish what it is. In linear algebra, the adjugate matrix, often denoted as \( \text{adj}(A) \), is the transpose of the cofactor matrix of a given square matrix \( A \). It essentially plays a pivotal role in finding the inverse of a matrix, especially when matrices are non-singular.

For a 3x3 matrix \( A \), the adjugate involves calculating the minors, co-factors, and then transposing the resultant matrix of co-factors. Imagine each entry in a matrix having a minor, a determinant of a 2x2 submatrix, calculated by excluding the row and column of that entry. The co-factors involve alternating signs and forming the cofactor matrix that gets transposed to become the adjugate.

Here's how you go about it:

• Find each minor by excluding the current row and column of the element.
• Apply a sign pattern, alternating between positive and negative.
• Transpose it to get the adjugate matrix.

This adjugate matrix links closely to the formula for calculating a matrix's inverse if it exists: \( A^{-1} = \frac{1}{|A|} \text{ adj}(A) \). It requires the determinant \(|A|\) to be non-zero, meaning the matrix is invertible. In essence, understanding the adjugate gives you the inverse's core building blocks.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, used to simplify linear transformations and solve systems of linear equations. This operation involves two matrices: say \( A \) and \( B \). It's crucial to note that matrix multiplication is not commutative, meaning \( AB \) does not equal \( BA \) in general.

The process of multiplying matrices involves pairing each row element of the first matrix with each column element of the second matrix, then summing their products to form the resulting matrix. Here's how:

• Check if the matrices can be multiplied by ensuring the number of columns in the first matrix equals the number of rows in the second.
• For each entry in the resulting matrix, pair the elements of the row from the first matrix and column of the second matrix, multiply them, and sum the results.
• This new matrix forms dimensions that reflect the rows of the first and columns of the second matrix.

This operation not only transforms and combines matrices but allows the creation of distinctly new matrices with properties that reflect transformations dependent on \( A \) and \( B \). Recognizing the role of sequential matrix products in transformations can aid in visualizing multidimensional data and managing systems efficiently.

Matrix Scalar Multiplication

Matrix scalar multiplication is one of the simplest operations in matrix algebra. This operation involves multiplying every entry in a matrix by the same scalar value. It adjusts the magnitude of all the matrix elements uniformly without altering the structure or the order of the matrix.

Consider a matrix \( A \) and a scalar \( k \). The scalar multiplication would result in a new matrix \( kA \), where each element of \( A \) is multiplied by \( k \). Here's a step-by-step of how to proceed:

• Pick a scalar, such as \( 3 \) in the context of the given problem.
• Multiply every entry in the matrix \( A \) by the scalar value \( k \).
• Note that the dimensions of the matrix remain unchanged, maintaining its original structure.

This operation signifies how certain transformations, like scaling, can be performed across entire matrix datasets, allowing for complex calculations like those seen in determinants that affect results on a global scale dependent on matrix dimensions. Understanding how scalars influence matrix results can simplify predicting outcomes and solving matrix-based problems.