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 Signup and view all the answers

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

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

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

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

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

(adjA)^m Signup and view all the answers

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

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

What is the adjugate of the identity matrix In?

In Signup and view all the answers

What is adj(0)?

0 Signup and view all the answers

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

adjA is also symmetric Signup and view all the answers

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

adjA is also diagonal Signup and view all the answers

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

adjA is also triangular Signup and view all the answers

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

|adjA|= 0 Signup and view all the answers

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

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 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) 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 Signup and view all the answers

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

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

Which of the following is generally true about matrix multiplication?

AB ≠ BA Signup and view all the answers

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

Identity or diagonal matrices Signup and view all the answers

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

Distributivity Signup and view all the answers

The associativity property of matrix multiplication implies:

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

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

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

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.

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Description

Test your knowledge on matrix operations, properties, and inverses. This quiz covers key concepts, including the adjugate of matrices and their determinants. Challenge yourself to solve problems related to matrix inverses and products.