Let A be a 2 ×2 matrix Statement-1: adj(adj A)=A Statement-2: |adj A|=|A| (A) Statement-1 is tru

step 1 In problem 36 we have a 2x2 matrix and we have two statement statement 1 says that the adjoint o

Let A be a 2 ×2 matrix Statement-1: adj(adj A)=A...

Let $A$ be a $2 \times 2$ matrix Statement-1: $\operatorname{adj}(\operatorname{adj} A)=A$ Statement-2: $|\operatorname{adj} A|=|A|$ (A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is true, Statement-2 is true; Statement- 2 is not a correct explanation for Statement-1 (C) Statement-1 is true, Statement- 2 is false (D) Statement-1 is false, Statement- 2 is true

Let $A$ be a $2 \times 2$ matrix Statement-1: $\operatorname{adj}(\operatorname{adj} A)=A$
Statement-2: $|\operatorname{adj} A|=|A|$
(A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is true, Statement-2 is true; Statement- 2 is not a correct explanation for Statement-1
(C) Statement-1 is true, Statement- 2 is false
(D) Statement-1 is false, Statement- 2 is true

Key Concepts

Adjugate Matrix

The adjugate matrix (or classical adjoint) of a square matrix is defined as the transpose of its cofactor matrix. It plays a vital role in expressing the inverse of a matrix; specifically, when the determinant of the matrix is nonzero, the inverse is given by the adjugate divided by the determinant. This concept is fundamental in linear algebra for solving systems of equations and understanding matrix inverses.

Determinant and Its Properties

The determinant is a scalar value associated with a square matrix that encapsulates key properties like invertibility and volume scaling under the linear transformation represented by the matrix. One of its key roles is in the relation A · (adj A) = |A|I, which links the matrix, its adjugate, and its determinant. The determinant’s multiplicative and power properties are essential in many matrix calculations, including those involving the adjugate.

Properties of Adjugate in 2x2 Matrices

In the special case of 2x2 matrices, there are neat relationships where applying the adjugate operator twice returns the original matrix, i.e., adj(adj A) = A, and the determinant of the adjugate equals the determinant of the original matrix, i.e., |adj A| = |A|. These properties arise due to the dimensional specificity of 2x2 matrices and are derived from general formulas that relate the determinant and the adjugate for n×n matrices.

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