Matrices Flashcards
What is a matrix?
A matrix is a rectangular array of numbers arranged in rows and columns.
How is a matrix typically denoted?
A matrix is typically denoted by a capital letter, such as A, B, or C.
What is the order of a matrix?
The order of a matrix is defined by the number of rows and columns it has, expressed as ‘m x n’ where m is the number of rows and n is the number of columns.
True or False: A matrix can have different numbers of elements in each row.
False.
Fill in the blank: A matrix with only one row is called a _______.
row matrix.
Fill in the blank: A matrix with only one column is called a _______.
column matrix.
What is a square matrix?
A square matrix is a matrix that has the same number of rows and columns.
Define a zero matrix.
A zero matrix is a matrix in which all elements are zero.
What is the identity matrix?
The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere.
True or False: The determinant of a 2x2 matrix can be calculated as ad - bc for a matrix [[a, b], [c, d]].
True.
What is the transpose of a matrix?
The transpose of a matrix is obtained by swapping its rows and columns.
How do you denote the transpose of a matrix A?
The transpose of a matrix A is denoted as A^T.
What is matrix addition?
Matrix addition is the operation of adding two matrices by adding their corresponding elements.
True or False: Matrices can only be added if they have the same dimensions.
True.
What is scalar multiplication in matrices?
Scalar multiplication involves multiplying each element of a matrix by a scalar (a single number).
What is matrix multiplication?
Matrix multiplication is the operation of multiplying two matrices by taking the dot product of rows and columns.
True or False: The order of multiplication matters in matrix multiplication.
True.
What is a diagonal matrix?
A diagonal matrix is a square matrix where all elements outside the main diagonal are zero.
What is an upper triangular matrix?
An upper triangular matrix is a square matrix where all the elements below the main diagonal are zero.
What is a lower triangular matrix?
A lower triangular matrix is a square matrix where all the elements above the main diagonal are zero.
Define an invertible matrix.
An invertible matrix, or non-singular matrix, is a square matrix that has an inverse, meaning there exists another matrix such that their product is the identity matrix.
What is the rank of a matrix?
The rank of a matrix is the maximum number of linearly independent row or column vectors in the matrix.
True or False: The rank of a matrix can be greater than the number of its rows.
False.
What is a singular matrix?
A singular matrix is a square matrix that does not have an inverse.
Fill in the blank: The process of finding the inverse of a matrix is called _______.
matrix inversion.
What is a block matrix?
A block matrix is a matrix that is divided into smaller matrices called blocks.
Who introduced the term matric8
It was introduced by the mathematician arther cayley in 1860
What are Rows denoted by
m
What are columns denoted by
n
When are two matrices equal
Two matrices are equal when they have the same order ie the same number of rows and columns and the elements in the corresponding positions are equal
What is a diagonal matrix?
A diagonal matrix is a square matrix in which all the elements outside the main diagonal are zero.
True or False: A scalar matrix is a special type of diagonal matrix.
True
Fill in the blank: An identity matrix is a diagonal matrix where all the diagonal elements are _____ .
1
What is the transpose of a matrix?
The transpose of a matrix is obtained by interchanging mutually the rows and columns.
True or False: A symmetric matrix is equal to its transpose.
True
What is the main characteristic of a scalar matrix?
A scalar matrix is a diagonal matrix where all the diagonal elements are equal.
Multiple Choice: Which of the following is an identity matrix? A) [[1, 0], [0, 1]] B) [[1, 2], [3, 4]] C) [[0, 1], [0, 0]]
A) [[1, 0], [0, 1]]
What is the result of transposing a 1x1 matrix?
The result is the same 1x1 matrix.
Fill in the blank: A symmetric matrix must satisfy the condition _____ .
A = A^T
True or False: All diagonal matrices are symmetric.
True
What happens when you multiply a scalar matrix by a scalar?
The result is another scalar matrix where each diagonal element is multiplied by that scalar.
Multiple Choice: Which of the following matrices is not a diagonal matrix? A) [[3, 0], [0, 4]] B) [[0, 1], [0, 0]] C) [[5, 0], [0, 5]]
B) [[0, 1], [0, 0]]
What is the size of an identity matrix of order n?
n x n
Fill in the blank: The transpose of a matrix A is denoted as _____ .
A^T
True or False: A matrix can be both diagonal and symmetric.
True
What is a skew symmetric matrix?
A skew symmetric matrix is a square matrix A such that A^T = -A, where A^T is the transpose of A.
True or False: All diagonal elements of a skew symmetric matrix are zero.
True
Fill in the blank: If A is a skew symmetric matrix, then for any elements a_{ij}, it holds that a_{ij} = _____ .
−a_{ji}
Which of the following matrices is skew symmetric? A) [[0, 2], [-2, 0]] B) [[1, 0], [0, 1]] C) [[1, 2], [3, 4]]
A) [[0, 2], [-2, 0]]
What is the property of the eigenvalues of a skew symmetric matrix?
The eigenvalues of a skew symmetric matrix are either zero or purely imaginary.
What is the condition for two matrices A and B to be conformable for multiplication AB?
The number of columns of matrix A must equal the number of rows of matrix B.
What does the associative law of matrices state for three matrices A, B, and C?
A(BC) = (AB)C.
True or False: The commutative law of multiplication holds for matrices in general.
False.
What is the left distributive law of matrices?
A(B + C) = AB + AC.
What is the right distributive law of matrices?
(A + B)C = AC + BC.
What is a determinant of a square matrix A denoted as?
det A or |A|.
What does it mean for a square matrix A to be singular?
It means |A| = 0.
What does it mean for a square matrix A to be non-singular?
|A| ≠ 0.
What is the adjoint of a square matrix A denoted as?
adj A.
If matrices A and B satisfy AB = BA = I, what is B called?
B is called the inverse of A.
If B is the inverse of A, how is it denoted?
B = A⁻¹.
Fill in the blank: A square matrix A is called ______ if |A| = 0.
singular.
Fill in the blank: A square matrix A is called ______ if |A| ≠ 0.
non-singular.
What is a matrix?
A matrix is a rectangular array of real numbers enclosed in brackets.
How do the rows and columns of a matrix run?
The rows of a matrix run horizontally, and the columns of a matrix run vertically.
What is the order of a matrix with m rows and n columns?
A matrix with m rows and n columns has order m X n (read ‘m by n’).
What is each number in a matrix called?
Each number in a matrix is called an element or entry of the matrix.
When are two matrices considered equal?
Two matrices of the same order are equal if their corresponding elements are equal.
What is a row matrix?
A matrix is said to be a row matrix if it has only one row.
What is a column matrix?
A matrix is said to be a column matrix if it has only one column.
What is a square matrix?
A matrix in which the number of rows is equal to the number of columns is called a square matrix.
What is a rectangular matrix?
A matrix in which the number of rows is not equal to the number of columns is called a rectangular matrix.
What is a zero matrix?
A matrix is said to be a zero matrix or null matrix if all its elements are zero.
What is a diagonal matrix?
If all the elements of a square matrix except the diagonal elements are zero, then the matrix is called a diagonal matrix.
What is a scalar matrix?
A diagonal matrix, whose all the diagonal elements are equal, is called a scalar matrix.
What is an identity matrix?
A scalar matrix of order n in which each diagonal element is 1 (unity) is called an identity matrix of order n.
What is the transpose of a matrix?
A matrix obtained by interchanging rows and columns of a matrix A is called the transpose of the matrix A and is denoted by A’.
What is a symmetric matrix?
If a square matrix A = A’, then A is called a symmetric matrix.
What is a skew-symmetric matrix?
A square matrix A is said to be skew-symmetric if A’ = -A.
What is the rule for the transpose of the sum of matrices?
(A + B)’ = A’ + B.
What is the rule for the transpose of the difference of matrices?
(A - B)’ = A’ - B’.
What does it mean for two matrices to be conformable for addition/subtraction?
Two matrices are said to be conformable for addition/subtraction if they are of the same order.
What is the null matrix in terms of matrix addition?
The null matrix is the additive identity for matrix addition.
What is the additive inverse of a matrix A?
If A is a matrix, then -A is the additive inverse of A.
