Problem 1 - Algebra for JEE ADVANCED

Question

Which of the following has/have value equal to zero? a. \(\left|\begin{array}{ccc}8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3\end{array}\right|\) b. \(\left|\begin{array}{ccc}1 / a & a^{2} & b c \\ 1 / b & b^{2} & a c \\ 1 / c & c^{2} & a b\end{array}\right|\) c. \(\left|\begin{array}{ccc}a+b & 2 a+b & 3 a+b \\ 2 a+b & 3 a+b & 4 a+b \\\ 4 a+b & 5 a+b & 6 a+b\end{array}\right|\) d. \(\left|\begin{array}{lll}2 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2\end{array}\right|\)

Step-by-Step Solution

Answer

Matrices b and c have determinants equal to zero.

Step 1: Evaluate Determinant a

Let's find the determinant of matrix a. The matrix is \[\begin{bmatrix} 8 & 2 & 7 \ 12 & 3 & 5 \ 16 & 4 & 3 \end{bmatrix}\].The determinant is calculated using the formula \( \text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg) \).For matrix a:- \( a = 8, b = 2, c = 7 \)- \( d = 12, e = 3, f = 5 \)- \( g = 16, h = 4, i = 3 \)Substitute these values into the formula:\[ \text{det}(A) = 8(3 \cdot 3 - 5 \cdot 4) - 2(12 \cdot 3 - 16 \cdot 5) + 7(12 \cdot 4 - 16 \cdot 3) \].Calculating each part:\[= 8(-11) - 2(-36) + 7(0) \].\[= -88 + 72 + 0 \].\[= -16 \].Therefore, determinant of matrix a is \(-16\), which is not zero.

Step 2: Evaluate Determinant b

Next, let's evaluate determinant b. The matrix is \[\begin{bmatrix} 1/a & a^2 & bc \ 1/b & b^2 & ac \ 1/c & c^2 & ab \end{bmatrix}\].Notice that the matrix can be manipulated into dependent rows, which often leads to a determinant of zero.Consider the method of cofactor expansion along the first row:For simplicity, you can swap rows or make row modifications:- Add the first row to the second and the third row, which simplifies the matrix.Each row becomes a linear combination of the others due to the specific construction:The rows can be seen as: \([\frac{1}{x}, x^2, yz] \) where \((x,y,z) = (a), (b), (c)\).Using this observation and trial cofactor simplification, we see the determinant is thus zero due to linear dependence.

Step 3: Evaluate Determinant c

Now, evaluate determinant c. The matrix is \[\begin{bmatrix} a+b & 2a+b & 3a+b \ 2a+b & 3a+b & 4a+b \ 4a+b & 5a+b & 6a+b \end{bmatrix}\].Observe a common pattern: each row is a linear combination of \(a\) and \(b\). Let's simplify:Each row can be written as \([a, 2a, 3a] + [b, b, b]\), implying that rows are not linearly independent.The typical approach is to subtract the first row from the others or reduce the matrix:Row operations show linear dependence, i.e., C is a conglomerate of vectors with the same coefficients.Perform row reductions:This results in determinants equating to zero due to linear dependence.

Step 4: Evaluate Determinant d

Lastly, let's evaluate determinant d. The matrix is \[\begin{bmatrix} 2 & 43 & 6 \ 7 & 35 & 4 \ 3 & 17 & 2 \end{bmatrix}\].Use the determinant formula:\( \text{det}(D) = a(ei - fh) - b(di - fg) + c(dh - eg) \).For matrix d:- \( a = 2, b = 43, c = 6 \)- \( d = 7, e = 35, f = 4 \)- \( g = 3, h = 17, i = 2 \)Substitute these values:\[ \text{det}(D) = 2(35 \cdot 2 - 17 \cdot 4) - 43(7 \cdot 2 - 3 \cdot 4) + 6(7 \cdot 17 - 3 \cdot 35) \].Calculate step by step:\[= 2(70 - 68) - 43(14 - 12) + 6(119 - 105) \].\[= 2(2) - 43(2) + 6(14) \].\[= 4 - 86 + 84 \].\[= 2 \].Determinant of matrix d is 2, which is not zero.

Key Concepts

Matrix Operations

Matrix operations are crucial in linear algebra because they allow us to analyze and manipulate matrices efficiently. Determinants, a specific type of matrix operation, provide valuable insight into the properties of square matrices, including their invertibility.

A determinant is a scalar value that can be computed from a square matrix and is denoted by vertical bars, like \( |A| \) for a matrix \( A \). Determinants play a key role in various applications:

• Solving linear systems: If the determinant of the coefficient matrix in a system of linear equations is zero, the system does not have a unique solution.
• Understanding matrix invertibility: A matrix is invertible if its determinant is non-zero.
• Transformations in geometry: Determinants can describe the scaling factor of linear transformations represented by matrices.

When performing matrix operations, one might often need to compute the determinant using methods like the cofactor expansion. This involves systematic calculation breaking down the matrix into smaller parts, enabling the evaluation of singular matrices (those with determinant zero) and identifying linear dependence among the rows or columns.

Linear Dependence

Linear dependence is an important concept in understanding the structure and properties of a matrix. Simply put, a set of vectors (or matrix rows/columns) is linear dependent if one of the vectors in the set can be expressed as a linear combination of others.

In the context of determinants, if the rows or columns of a matrix are linearly dependent, then the determinant of the matrix is zero, indicating the matrix is singular and not invertible. This concept is crucial when evaluating determinants, especially for matrices arising from theoretical or practical problems.

• Zero determinant: If any row or column of a matrix is a linear combination of other rows or columns, the determinant is zero.
• Unique solutions in systems: In linear equations, a zero determinant implies non-unique solution set.
• Rank of a matrix: Linear dependence affects the rank, which is the maximum number of linearly independent vectors in the matrix.

Identifying linear dependence can significantly simplify determinant calculation, and it often involves recognizing patterns or performing row operations to reveal dependence among the vectors.

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a matrix. This technique is versatile and works with any square matrix, regardless of size. It involves expanding the determinant across a particular row or column, multiplying each element by its cofactor, and then summing up the results.

The cofactor of an element in a matrix is calculated by taking the determinant of the minor matrix, obtained by removing the row and column of the element, and then applying a sign change according to its position.

• Alternating signs: The sign is determined by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices.
• Recursive method: Cofactor expansion can be applied recursively to smaller matrices until reaching a trivial case, typically a 2x2 or 1x1 matrix, making it manageable even for larger matrices.

Cofactor expansion is especially useful to calculate determinants of higher-order matrices where direct computation would be tedious. Additionally, understanding this method aids in recognizing the role of linear dependence since linearly dependent rows lead to zero determinants through cofactor calculations.