Linear Algebra: Matrices

Question 1

Linear System

Answer 1

A finite set of linear equations in the variables x1, x2, … xn

Question 2

Inconsistent linear system

Answer 2

A system with no solutions.

Question 3

Consistent linear system

Answer 3

A system with at least one solution.

Question 4

3 Elementary row operations

Answer 4

• Multiply a row throughout by a nonzero constant - Interchange two rows - Add a multiple of one row to another

Question 5

Row Echelon form

Answer 5

A matrix with properties: - The first nonzero number in every row is a 1 (leading 1). (Except if a row consists entirely of zeros). - Any rows consisting entirely of zeros are grouped together at the bottom of the matrix. - The leading 1 in the lower row occurs further to the right than the leading row in the higher row (in non-zero rows).

Question 6

Reduced Row Echelon form

Answer 6

A matrix with all the properties of Row Echelon form and: - Each column that has a leading 1 has zeros everywhere else in that column.

Question 7

General Solution

Answer 7

A set of parametric equations from which all solutions of a linear system can be obtained by assigning numerical values to the parameters.

Question 8

Homogeneous

Answer 8

A system of linear equations where the constant terms are all zero.

Question 9

Trivial solution to a homogeneous system

Answer 9

x1 = x2 = …. = xn = 0

Question 10

Nontrivial solution to a homogeneous system

Answer 10

Any solution other than the trivial solution

Question 11

2 Possibilities for any homogenous linear system:

Answer 11

• Only the trivial solution - Infinitely many solutions, including the trivial solution.

Question 12

Number of solutions to a homogenous system with more unknowns than equations.

Answer 12

Infinite

Question 13

Number of free variables in a homogenous linear system that has n unknowns, and a reduced row echelon form with r nonzero rows.

Answer 13

n-r

Question 14

Pivot positions

Answer 14

The positions of the leading 1’s.

Question 15

Matrix

Answer 15

A rectangular array of numbers - called entries.

Question 16

Column matrix

Answer 16

A matrix with only one column

Question 17

Row matrix

Answer 17

A matrix with only one row

Question 18

Scalars

Answer 18

Numerical quantities

Question 19

Equal matrices

Answer 19

Matrices with the same size, whose corresponding entries are equal.

Question 20

Transpose of a matrix

Answer 20

A matrix of the same size as the original, obtained by interchanging the rows and columns of A.

Question 21

Trace

Answer 21

The trace of a square matrix, denoted tr(A), is the sum of the entries on the main diagonal of the matrix.

Question 22

Zero matrix

Answer 22

Any matrix with only zero entries.

Question 23

Identity matrix

Answer 23

Square matrix with 1’s on the main diagonal and zero’s elsewhere.

Question 24

Explain an inverse

Answer 24

If A is a square matrix, and B a matrix of the same size, such that AB = BA = I, then A is said to be invertible and B is called an inverse of A.

Question 25

Singular matrix

Answer 25

A matrix that has no inverse.

Question 26

Nonnegative integer powers of a matrix (A)

Answer 26

A⁰ = I

Aⁿ = AA … A (n times, n>0)

Question 27

Negative integer powers of a matrix (A)

Answer 27

A^-n = ( A^-1)ⁿ = A^-1A^-1… A^-1 (n times)

Question 28

Exponential Laws of matrices

Answer 28

If A is a square matrix and r and s are integers then:

• A^rA^s = A^r+s
• ( A^r)^s = A^rs

Question 29

Define constant a in matrix terms

Answer 29

a.I

Question 30

5 Properties of the transpose of a matrix

Answer 30

• ( (A)^T)^T = A
• ( A+B )^T = A^T + B^T
• ( A-B )^T = A^T - B^T
• ( kA )^T = kA^t, k being any scalar
• ( AB )^T = B^TA^T

Question 31

Elementary matrix

Answer 31

A matrix that can be obtained from the nxn identity matrix In by performing a single elementary row operation.

Question 32

Theorem 1.5.3 on matrices (4 all true/all false)

Answer 32

• A is invertible
• A_x = 0 has only the trivial solution
• The reduced row echelon form of A is I_n
• A is expressible as a product of elementary matrices

Question 33

Row equivalent matrices

Answer 33

Matrices that can be obtained from one another by a finite sequence of elementary row operations.

Question 34

Method for inverting matrices

Answer 34

We must find a sequence of elementary row operations that reduces A to the identity and then perofrm this same sequence of operations to I_n to obtain A^-1.