Matrix Algebra Class 10

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Questions and Answers

What is the order of a square matrix?

n × n where n is any positive integer (correct)

n × m where n > m

m × n where m = n

m × n where m ≠ n

Which of the following matrices is termed a zero matrix?

A matrix with at least one non-zero element

A matrix with all elements equal to one

A matrix with alternating zero and non-zero elements

A matrix with all elements being zero (correct)

What is the defining characteristic of an identity matrix?

It can have any shape and still be an identity matrix

It has diagonal elements equal to one and all other elements equal to zero (correct)

It contains all zero elements

It is always a 2 × 2 matrix

What is the size of the given zero matrix S?

2 × 2 Signup and view all the answers

If A is a 3 × 3 identity matrix, what will be the result of multiplying A by any matrix B of size 3 × n?

It will result in a matrix of size 3 × n Signup and view all the answers

Which of these statements is true regarding square matrices?

Square matrices are defined by having equal number of rows and columns Signup and view all the answers

Which of the following elements is NOT found in any square matrix?

Non-square elements Signup and view all the answers

How can an identity matrix be denoted?

I Signup and view all the answers

What size will the resulting matrix AB have if A is m × p and B is p × n?

m × n Signup and view all the answers

What is the formula for a specific entry in matrix AB at position (1, 1)?

ae + bg Signup and view all the answers

Which of the following correctly demonstrates the non-commutative property of matrix multiplication?

AB != BA Signup and view all the answers

If A is a 2 × 3 matrix and B is a 3 × 2 matrix, what is the size of the product AB?

2 × 2 Signup and view all the answers

Which entry of matrix AB corresponds to the sum of products from row 1 of A and column 2 of B?

af + bh Signup and view all the answers

In the given matrices A and B, what does the entry at position (2, 1) of the product AB represent?

ce + dg Signup and view all the answers

What is the result of multiplying two matrices A and B if the number of columns of A does not match the number of rows of B?

The multiplication is undefined Signup and view all the answers

If A is defined as a matrix where all entries are 1, what will be the sum of the entries in the resulting matrix AB if B has dimensions 3 × 2?

6 Signup and view all the answers

What is the trace of a matrix A defined as?

The sum of the main diagonal elements Signup and view all the answers

If matrix A is given as ( A = \begin{pmatrix} 1 & 2 \ 3 & 4 \ \end{pmatrix} ), what is trace(A)?

5 Signup and view all the answers

Which property of trace states that the trace of the sum of two matrices equals the sum of their traces?

trace(A + B) = trace(A) + trace(B) Signup and view all the answers

If ( c ) is a scalar, what does the property ( trace(cA) ) equal?

c \cdot trace(A) Signup and view all the answers

What is the value of ( trace(A^2) ) if ( A ) is a 2x2 identity matrix?

2 Signup and view all the answers

What is the relationship expressed by the property ( trace(AB) = trace(BA) )?

Trace is invariant under cyclic permutations Signup and view all the answers

For which condition the property trace(cA) holds true?

For any size matrix Signup and view all the answers

What is the identity matrix ( I ) of size 2x2?

( \begin{pmatrix} 1 & 0 \ 0 & 1 \ \end{pmatrix} ) Signup and view all the answers

What is the result of the matrix addition A+B if A is given as $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and B as $\begin{bmatrix} 5 & 6 \ 0 & -1 \end{bmatrix}$?

$\begin{bmatrix} 6 & 8 \ 3 & 3 \end{bmatrix}$ Signup and view all the answers

Which property of matrix addition is represented by A + (−A) = 0?

Additive inverse property Signup and view all the answers

If tA = 0 implies t = 0 or A = 0, which property does this represent?

Zero product property Signup and view all the answers

What is the shape of the resulting matrix when multiplying an m × p matrix A with a p × n matrix B?

m × n Signup and view all the answers

Which of the following is NOT a property of matrix addition?

0 + 0 = 0 Signup and view all the answers

What is the result of scalar multiplication 2B if B is $\begin{bmatrix} 5 & 6 \ 0 & -1 \end{bmatrix}$?

$\begin{bmatrix} 10 & 12 \ 0 & -2 \end{bmatrix}$ Signup and view all the answers

Which of these statements regarding matrix subtraction is true?

A - B = A + (−B). Signup and view all the answers

Which property applies when a scalar t is multiplied by the sum of two matrices?

t(A + B) = tA + tB Signup and view all the answers

What condition must a matrix A satisfy to be classified as symmetric?

$A^T = A$ Signup and view all the answers

Which of the following matrices is skew symmetric?

$egin{pmatrix} 0 & -3 \ 3 & 0 \ \ \ \ \ \ \ \ \ \ \end{pmatrix}$ Signup and view all the answers

What is the primary characteristic of a symmetric positive definite matrix?

It yields positive results for all nonzero vectors when used in the quadratic form. Signup and view all the answers

Which of the following statements is true about skew symmetric matrices?

Their diagonal elements must be zero. Signup and view all the answers

What happens to the elements of a skew symmetric matrix when you take the transpose?

The sign of all elements changes. Signup and view all the answers

Which of the following is not a property of symmetric matrices?

The eigenvalues can be both positive and negative. Signup and view all the answers

If matrix A is symmetric, what can be said about its eigenvalues?

They are always real numbers. Signup and view all the answers

What does the notation $A^T$ signify in matrix operations?

The transpose of matrix A. Signup and view all the answers

What is the condition for a matrix A to be classified as Hermitian?

ĀT = A Signup and view all the answers

Which of the following matrices is skew Hermitian?

N =  -2+i & 3i & -3i  Signup and view all the answers

In which case will the matrix C defined as C =  2 & 1  be defined?

When C is multiplied by any matrix of compatible dimensions. Signup and view all the answers

What would be the result of multiplying matrices A and B where A =  3 & 0  and B =  -1 & 1 & 0 ?

Result is undefined due to incompatible dimensions. Signup and view all the answers

Identify the correct representation of a Hermitian matrix.

A = ĀT Signup and view all the answers

Which of the following statements about a skew Hermitian matrix is true?

Their diagonal entries are purely imaginary. Signup and view all the answers

If matrices A and D are defined, which operation can be performed without any limitations?

D², since squaring requires the matrix to be square. Signup and view all the answers

What is the primary characteristic of the matrix xσ1 + yσ2 + 2σ3?

It can only be Hermitian if x and y are both real. Signup and view all the answers

Study Notes

Introduction to Matrices

Matrices are rectangular arrays of numbers, symbols, or properties of mathematical objects.
They represent linear maps.
The notation R_mxn represents the collection of all m x n matrices whose entries are real numbers.

Definition of Matrices

A matrix is a rectangular array with m rows and n columns.
Matrices are usually denoted by capital letters.
The notation A = [a_ij] indicates that the matrix is composed of entries a_ij, located in the i^th row and j^th column of A.

Types of Matrices

Square Matrices: Matrices with the same number of rows and columns (m x n)

Zero Matrix (0)

A matrix with all elements equal to zero (size m x n).

Identity Matrix (I)

A square matrix (n x n) where all diagonal elements are 1, and all other elements are 0.
The identity matrix acts like the number 1 in standard arithmetic.
It is also known as the unit matrix.

Equality of Matrices

Two matrices A and B are equal if they have the same size and all their corresponding elements are equal.

Matrix Arithmetic Operations: Addition

Given two matrices A and B of the same size (m x n), A + B is the matrix obtained by adding the corresponding elements.
A + B = [a_ij] + [b_ij] = [a_ij + b_ij]

Matrix Arithmetic Operations: Scalar Multiplication

Given a matrix A = [a_ij] and a scalar t, tA is the matrix obtained by multiplying all elements of A by t.
tA = t[a_ij] = [ta_ij]

Matrix Arithmetic Operations: Negative of a Matrix

The negative of a matrix A is -A, obtained by replacing each element of A with its negative.

Matrix Arithmetic Operations: Subtraction

For two matrices A and B of the same size, A - B is defined as subtracting the corresponding elements.
- A - B = [a_ij] - [b_ij] = [a_ij - b_ij]

Matrix Product

The product AB is an m x n matrix.
The number of columns of A must equal the number of rows of B.

Matrix Multiplication Example

Shows how to multiply two matrices and obtain the result.

The Trace of a Matrix

For an n x n matrix A, the trace of A (trace(A)) is the sum of diagonal elements.
- trace(A) = a₁₁ + a₂₂ + ... + a_nn = Σa_ii

Properties of the Trace Operations

Trace(A + B) = Trace(A) + Trace(B), Trace(cA) = c * Trace(A) and Trace(AB) = Trace(BA)

Power of a Matrix

A^k is defined as multiplying a matrix A by itself k times. A⁰ = I (identity matrix)

Transpose of a Matrix

The transpose of a matrix A (denoted as A^T) is obtained by interchanging rows and columns of A.

Properties of Transposes

(A^T)^T = A
(A + B)^T = A^T + B^T
(sA)^T = sA^T , where s is a scalar.
(AB)^T = B^TA^T if A is m x k and B is k x n

Diagonal Matrix

A square matrix where only the diagonal elements are nonzero.

Bidiagonal Matrix

A square matrix with nonzero elements only on the main diagonal and the adjacent diagonals above or below.

Tridiagonal Matrix

A square matrix with nonzero entries along the main diagonal and the two adjacent diagonals above and below the main diagonal.

Nonsingular Matrix

A square matrix that has an inverse.
An invertible matrix is a nonsingular matrix.

Properties of the Inverse

(A^-1)A = I = A(A^-1)
(A^-1)^-1 = A
(AB)^-1 = B^-1A^-1, if A is nonsingular, A^T is also nonsingular and (A^T)^-1 = (A^-1)^T

Symmetric Matrix

A square matrix equal to its transpose (A = A^T).

Skew-Symmetric Matrix

A square matrix whose transpose is the negative of itself (A^T = −A).

Symmetric Positive Definite Matrix

A symmetric matrix for which, for every nonzero vector x, x^TAx > 0

Symmetric Positive Semidefinite Matrix

A symmetric matrix with x^TAx≥0

Symmetric Indefinite Matrix

A symmetric matrix that takes on both positive and negative values when x^TAx

Orthogonal Matrix

A square matrix whose transpose is also its inverse (P^TP = I)

Orthonormal

A set of orthogonal vectors with a unit length.
e₁, e₂, ... , e_n are the standard orthonormal basis.

Complex Conjugate Matrix

The complex conjugate of a matrix A, denoted by A, is obtained by taking the complex conjugate of each entry.

Hermitian Matrix

A complex square matrix A is hermitian if A^T = A (or Z_ji = Z_ij).

Skew-Hermitian Matrix

A complex square matrix A is skew-hermitian if A^T = -Ā (or Z_ji = -Z_ij).

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Description

Test your knowledge of matrix algebra with this quiz designed for Class 10 students. Questions cover topics such as identity matrices, zero matrices, and properties of square matrices. Determine your understanding of matrix multiplication and its properties.