Matrix Operations and Inverses Quiz

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Questions and Answers

If A is a square matrix, which of the following is true?

A = |A| In
A(adjA) = In
A(adjA) = (adjA)A
A(adjA) = |A| In = (adjA)A (correct)

What is the value of |adjA| for a square matrix A?

|A|^n
|A|^(-1)
|A|^2
|A|^(n-1) (correct)

For a square matrix A, what is the formula for adj(adjA)?

(adjA)^2
A^2
|A|^(n-2) A (correct)
|A| A

What is the formula for |adj(adjA)|?

|A|^(n-1)^2 (C) Signup and view all the answers

Which of the following is true for adjugates of transposed matrices?

adj(A^T) = (adjA)^T (A) Signup and view all the answers

For two square matrices A and B, which of the following is true?

adj(AB) = (adjB)(adjA) (A) Signup and view all the answers

For any natural number m, what is the formula for adj(A^m)?

(adjA)^m (D) Signup and view all the answers

If k ∈ ℝ and A is a square matrix, what is adj(kA)?

k^(n-1) ⋅ adjA (A) Signup and view all the answers

What is the adjugate of the identity matrix In?

In (C) Signup and view all the answers

What is adj(0)?

0 (D) Signup and view all the answers

If A is a symmetric matrix, which of the following is true about adjA?

adjA is also symmetric (A) Signup and view all the answers

If A is a diagonal matrix, which of the following is true about adjA?

adjA is also diagonal (A) Signup and view all the answers

If A is a triangular matrix, what is true about adjA?

adjA is also triangular (A) Signup and view all the answers

Which of the following is true if A is a singular matrix?

|adjA|= 0 (B) Signup and view all the answers

If A is an invertible matrix, what is (A^(-1))^(-1)?

A (A) Signup and view all the answers

Which of the following is true about the inverse of the transpose of a matrix?

(A^T)^(-1) = (A^(-1))^T (B) Signup and view all the answers

For two invertible matrices A and B, which of the following is true for their product?

(AB)^(-1) = B^(-1)A^(-1) (C) Signup and view all the answers

Which of the following is correct for the inverse of a matrix power A^k, where k ∈ ℕ?

(A^k)^(-1) = (A^(-1))^k (B) Signup and view all the answers

If A is an invertible matrix, what is adj(A^(-1))?

(adjA)^(-1) (D) Signup and view all the answers

For any invertible matrix A, which of the following is true about its determinant?

|A^(-1)|= 1/|A|=|A|^(-1) (D) Signup and view all the answers

If A is a diagonal matrix A = diag(a1, a2, …, an), what is A^(-1)?

A^(-1) = diag(a1^(-1), a2^(-1), …, an^(-1)) (D) Signup and view all the answers

Which of the following is generally true about matrix multiplication?

AB ≠ BA (D) Signup and view all the answers

Which of the following matrix types allows for commutative multiplication, i.e., AB = BA?

Identity or diagonal matrices (D) Signup and view all the answers

Which property of matrix multiplication states that A(B + C) = AB + AC?

Distributivity (D) Signup and view all the answers

The associativity property of matrix multiplication implies:

(AB)C = A(BC) (A) Signup and view all the answers

Which of the following expressions demonstrates the distributive property of matrix multiplication?

A(B + C) = AB + AC (C) Signup and view all the answers

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Study Notes

Matrix Operations and Properties

For a square matrix ( A ), ( A \cdot (\text{adj},A) = |A| I_n ) and also ( (\text{adj},A) \cdot A = |A| I_n ).
The relationship holds for adjugates of transposed matrices: ( \text{adj}(A^T) = (\text{adj},A)^T ).
Determinant of the adjugate of a matrix ( A ) is given by ( |\text{adj},A| = |A|^{n-1} ).
For the product of two matrices, ( \text{adj}(AB) = \text{adj},B \cdot \text{adj},A ) holds true.
The adjugate of a symmetric matrix remains symmetric: if ( A ) is symmetric, then ( \text{adj},A ) is also symmetric.

Inverse of Matrices

The inverse of an inverse matrix is the original matrix: ( (A^{-1})^{-1} = A ).
For the transpose of matrix ( A ), the inverse satisfies the equation ( (A^T)^{-1} = (A^{-1})^T ).
The inverse of the product of two matrices is given by ( (AB)^{-1} = B^{-1} A^{-1} ).
The inverse of a power of matrix ( A^k ) is expressed as ( (A^k)^{-1} = (A^{-1})^k ).

Determinants and Singular Matrices

The determinant of the inverse matrix is ( |A^{-1}| = \frac{1}{|A|} = |A|^{-1} ).
If ( A ) is a singular matrix, then ( |\text{adj},A| = 0 ).
For a diagonal matrix ( A = \text{diag}(a_1, a_2, ..., a_n) ), its inverse is ( A^{-1} = \text{diag}(a_1^{-1}, a_2^{-1}, ..., a_n^{-1}) ).

Matrix Multiplication Properties

Matrix multiplication is not commutative in general, ( AB \neq BA ).
When multiplying matrices, the associative property applies: ( (AB)C = A(BC) ).
The distributive property exists in matrix multiplication, ( A(B + C) = AB + AC ).
Commutative multiplication occurs primarily with identity or diagonal matrices.

Key Formulas

For any natural number ( m ), ( \text{adj}(A^m) = (\text{adj},A)^m ).
For a scalar ( k ), ( \text{adj}(kA) = k^{n-1} \cdot \text{adj},A ) for a square matrix ( A ).
The adjugate of the identity matrix ( I_n ) is the identity matrix itself.

Summary of Key Relationships

( \text{adj}(A) ) for an invertible ( A ): ( \text{adj}(A^{-1}) = (\text{adj},A)^{-1} ).
Properties of specific types of matrices (diagonal, symmetric, triangular) relate directly to their adjugates and inverses.