Matrices

Question 1

What is a row?

Answer 1

A horizontal arrangement of elements in a matrix.

Question 2

What is an element?

Answer 2

Each individual entry in a matrix.

Question 3

What is a column?

Answer 3

A vertical arrangement of elements in a matrix.

Question 4

What is a dimension?

Answer 4

The size of a matrix, given by the number of rows and columns. For example, a matrix with 3 rows and 2 columns has dimensions 3x2.

Question 5

What is a square matrix?

Answer 5

A matrix with the same number of rows and columns.

Question 6

What is an identity matrix?

Answer 6

A square matrix with ones on the main diagonal and zeros elsewhere.

Question 7

When can matrices be added?

Answer 7

Matrices can be added if they have the same dimensions.

Question 8

How do you add matrices?

Answer 8

Add corresponding elements.

Question 9

When can matrices be subtracted?

Answer 9

Matrices can be subtracted if they have the same dimensions.

Question 10

How do you add matrices?

Answer 10

Add corresponding elements.

Question 11

When can you multiply matrices?

Answer 11

Matrices can be multiplied if the number of columns in A equals the number of rows in B.

Question 12

What would be the resulting shape of matrix multiplication with dimensions 4x3 and 3x2?

Answer 12

The resulting shape of the matrix multiplication will be 4x2.

Question 13

What is the determinant of a 2x2 matrix, [a,b ; c,d]?

Answer 13

The determinant is given by ad - bc.

Question 14

What is the determinant of a 3x3 matrix, [a,b,c ; d,e,f ; g,h,i]?

Answer 14

The determinant is calculated using the formula:

det = a(ei - fh) - b(di - fg) + c(dh - eg)

Question 15

What is the determinant of the identity matrix?

Answer 15

The determinant of the identity matrix is 1.

Question 16

What can be said about the determinant of a matrix and its transpose?

Answer 16

The determinant of a matrix and its transpose are the same.

Question 17

How does multiplying a row or column of a matrix by a scalar affect its determinant?

Answer 17

Multiplying a row or column of a matrix by a scalar multiplies the determinant by that scalar.

Question 18

What happens to the determinant when two rows or columns of a matrix are interchanged?

Answer 18

Interchanging two rows or columns of a matrix changes the sign of the determinant.

Question 19

If two rows or columns of a matrix are proportional, what is the determinant?

Answer 19

If two rows or columns of a matrix are proportional, then the determinant is zero.

Question 20

What is the transpose of a matrix?

Answer 20

The transpose of a matrix is obtained by interchanging its rows and columns.

Question 21

If matrix A has dimensions m x n, what are the dimensions of its transpose A^T?

Answer 21

The transpose A^T will have dimensions n x m.

Question 22

What are the conditions for a square matrix to be symmetric?

Answer 22

A square matrix is symmetric if it is equal to its transpose, A = A^T.

Question 23

Assuming the products between the matrices P, Q, R exist, are the following relations always satisfied?

i) (P + Q)R = PR + QR
ii) PQ = QP

Answer 23

i) Yes
ii) No

Question 24

What is the determinant of a matrix?

Answer 24

The determinant of a square matrix is a scalar value that can be computed from its elements.

Question 25

Define minor in the context of matrices.

Answer 25

The minor of an element in a matrix is the determinant of the matrix obtained by deleting the row and column containing that element.

Question 26

What is a cofactor in the context of matrices?

Answer 26

The cofactor of an element in a matrix is the signed minor of that element.

Question 27

What is the adjugate matrix of a square matrix?

Answer 27

The adjugate matrix of a square matrix is obtained by taking the transpose of the matrix of cofactors of the original matrix.

Question 28

Define adjugate matrix.

Answer 28

The adjugate matrix of a square matrix is the transpose of the matrix of cofactors of the original matrix.

Question 29

How do you calculate the inverse of a matrix using the determinant?

Answer 29

A^−1 = ( 1 / det(A) ) ⋅ adj(A)

Question 30

What is the determinant of a square matrix that indicates whether it has an inverse?

Answer 30

The determinant of the matrix must be non-zero for it to have an inverse.

Question 31

True or False: Every square matrix has a unique inverse.

Answer 31

True. Every non-singular (invertible) square matrix has a unique inverse, or it is singular and does not have an inverse.

Question 32

How is the inverse of a product of matrices AB related to the inverses of the individual matrices A and
B?

Answer 32

he inverse of AB is equal to the product of the inverses of B and A in reverse order (AB)^(-1) = B^(-1) ⋅ A^-(1)

Question 33

What is the definition of a set of linear equations?

Answer 33

Sets of linear equations consist of multiple equations representing linear relationships between variables.

Question 34

How are linear equations represented using matrices?

Answer 34

Coefficient matrix (A), variable matrix (X), and constant matrix (B).

Question 35

Name a method for solving sets of linear equations that involves transforming the augmented matrix into row-echelon form.

Answer 35

Gaussian elimination.

Question 36

Define the terms “consistent” and “inconsistent” in the context of sets of linear equations.

Answer 36

A set of equations is consistent if it has at least one solution, and inconsistent if it has no solution.

Question 37

How many solutions can a set of linear equations have?

Answer 37

Zero, one, or infinitely many solutions.

Question 38

State a condition on the matrix A for the set of ‘n’ inhomogeneous equations in ‘n’ unknowns AX = b to have a unique solution.

Answer 38

• Determinant doesn’t equal 0
• Inverse matrix exists
• Matrix is non-singular

Question 39

State a condition on the matrix A for the set of n homogeneous equations in n unknowns AX = 0 to
have a non-trivial solution.

Answer 39

• Determinant equals 0
• Inverse matrix doesn’t exists
• Matrix is singular