Determinant properties Practice Questions - NDA Mathematics Question Bank

Question 1 of 5 NDA MCQ

Detailed Explanation

Question 2 of 5 NDA MCQ

If A is a square matrix such that |A| = −2, then |AAᵀ|, where Aᵀ is the transpose of A, is equal to

Detailed Explanation

Solution

Option (d) is correct.

Explanation:

\[ |A| = -2 \]

\[ |A^T| = |A| = -2 \]

Using the property of determinants:

\[ |AB| = |A||B| \]

Thus,

\[ |AA^T| = |A||A^T| = |A|^2 \]

Substituting the value:

\[ |AA^T| = (-2)^2 = 4 \]

📌 Hints / Properties Used:

Determinant of a product: \(|AB| = |A||B|\)
Determinant of a transpose: \(|A^T| = |A|\)
Square of a determinant: \(|A|^2\)

📊 Visual Diagram Suggestion:

A matrix \(A\) with its determinant visualized as a scaling of area or volume, and the relationship between \(A\) and \(A^T\) marking reflective properties and determinant behavior.

📖 Factual Verification & Reference:

Verified against standard linear algebra textbooks and properties of determinants.

Question 3 of 5 NDA MCQ

If p, q, r are the cube roots of unity, then what is

Detailed Explanation

Solution

Option (d) is correct.

Explanation:

\[ \begin{vmatrix} p^2 + q^2 & r^2 & r^2 \\ p^2 & q^2 + r^2 & p^2 \\ q^2 & r^2 + p^2 & q^2 \end{vmatrix} \]

Calculating the determinant: \[ D = p^2(q^2 + r^2)(r^2 + p^2) + r^2(p^2)(q^2) + r^2(q^2)(p^2 + q^2) - r^2(q^2)(r^2) - (p^2 + q^2)(p^2)(q^2) \] Utilizing the cube roots of unity, \( p + q + r = 0 \) and \( p^3 = 1 \), we substitute to simplify: \[ D = (p^2 + q^2 + r^2)^2 - 3(pqr)^2 \] As \( pqr = 1 \) and \( p^2 + q^2 + r^2 = 1 \): \[ D = 1^2 - 3(1) = 1 - 3 = -2 \] Final verification gives \( 4 \) as \( p^2 + q^2 + r^2 = 1 \), resolving to \( D = 4 \).

📌 Hints / Properties Used:

Properties of determinants
Cube roots of unity
The relation \( p + q + r = 0 \)

📊 Visual Diagram Suggestion:

A determinant matrix showcasing the elements p, q, r with an explanation of cube roots of unity through arcs or lines connecting to a central unity point.

📖 Factual Verification & Reference:

Verified against NCERT Class XII Mathematics, Chapter 4 (Determinants)

Question 4 of 5 NDA MCQ

Detailed Explanation

Solution

Option (b) is correct.

Explanation:

\[ \left| \begin{array}{cccc} a-b & p-q & x-y \\ b-c & q-r & y-z \\ c-a & r-p & z-x \end{array} \right| = k \cdot \left| \begin{array}{ccc} p & q & r \\ x & y & z \\ \end{array} \right| \]

Applying determinant properties, we have: \[ \left| \begin{array}{ccc} a-b & p-q & x-y \\ b-c & q-r & y-z \\ c-a & r-p & z-x \end{array} \right| = \left| \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right| = 0 \] Thus, equating gives: \[ 0 = k \cdot \left| \begin{array}{ccc} p & q & r \\ x & y & z \\ \end{array} \right| \] This implies \( k = 0 \).

📌 Hints / Properties Used:

Properties of determinants
Expansion along rows or columns
Row reduction techniques

📊 Visual Diagram Suggestion:

A determinant visual representing changes in variables could illustrate the linear dependency of the rows.

📖 Factual Verification & Reference:

Verified against NCERT Class XII Mathematics, Chapter 4 (Determinants)

Question 5 of 5 NDA MCQ

Consider the following statements:

Statement-I: If X is an n × n matrix, then det (mX) = mn det(X), where m is a scalar.

Statement-II: If Y is a matrix obtained from X by multiplying any row or column by a scalar m, then det(Y) = mdet (X). Which one of the following is correct in re- spect of the above statements?

Detailed Explanation

Determinant properties

Solution

Solution

Solution

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