Matrices

Question 1

Give a basic definition of a matrix.

Answer 1

A matrix is a set of real or complex numbers (or elements) arranged in rows and columns to form a rectangular array.

A matrix having m rows and n columns is called an m x n matrix, and is referred to as having order m x n

If m = n then the array is square, and the array is then called a square matrix of order n.

If the matrix has one column or one row it is then called a column vector or row vector respectively.

Question 2

What does the general entry a_ij represent in a matrix?

Answer 2

a_ij represents an element in a matrix that is positioned in the i^{<strong>th</strong>} row and the j^thcolumn.

The element can be real or complex.

Question 3

What is a diagonal matrix?

Answer 3

A diagonal matrix is a square matrix that has its only non-zero elements along the leading diagonal (it may have zeros on the leading diagonal also).

Question 4

Define the leading diagonal and the trace of a matrix.

Answer 4

In a square matrix of order n the diagonal containing the elements a₁₁, a₂₂, a₃₃, … , a_nn is called the principle, main, or leading diagonal.

The sum of the elements of the leading diagonal is called the trace of the square matrix.

e.g.

trace A = a₁₁ + a₂₂ + … + a_nn = Σ_{i = 1}ⁿ of a_ii

Question 5

Define the Transpose of a matrix.

Answer 5

The transposed matrix A^T is the matrix with elements b_ij = a_ji

In essence, it is a matrix with the rows and columns interchanged. see image below

Note: a column vector it transposed to a row vector and vice versa.

Question 6

What are the rules of matrix addition?

Answer 6

You can only add an m x n matrix with another m x n matrix.

An element of the sum matrix is the sum of the corresponding elements from the original matrices.

If A has elements a_ij and B has elements b_ij then A+B has elements a_ij + b_ij

Similarly for subtraction, A - B has elements a_ij - b_ij

Question 7

Define multiplication of a matrix by a scalar.

Answer 7

Question 8

What is a symmetrical matrix?

Answer 8

A Symmetrical Matrix is one, in which (a_ij) = (a_ji)

Meaning that the it is symmetric about the leading diagonal.

This also means that A = A^T for a symmetric matrix.

Question 9

What is a Skew-Symmetric Matrix?

Answer 9

A Skew-Symmetric Matrix is one in which (a_ij) = - (a_ji)

Meaning that the triangle of numbers below the leading diagonal are the negatives of the numbers above the leading diagonal and vice versa.

Note that this means that A = - A^T

Question 10

What is a Diagonal Matrix?

Answer 10

A Diagonal Matrix is one in which all of the elements of the matrix are zero except those on the leading diagonal.

Question 11

What is a Null Matrix?

Answer 11

In a Null Matrix all of the elements of the matrix are zero.

Question 12

What is a Singular Matrix?

Answer 12

A Singular Matrix is one whose determinant is zero.

Question 13

What is the relationship between det(A) and det(A^T)?

Use the following matrix to figure it out:

( 1 2 )

( 3 4 )

Answer 13

the determinant of the matrix A is equal to the determinant of the transpose of matrix A.

i.e.

det(A) = det(A^T)

Question 14

What is the result of muliplying a matrix with its inverse?

Answer 14

When you multiply a matrix with its inverse (in either order) the result is the Identity Matrix.

i.e.

A.A^{- 1} = A^{- 1}.A = I_m

Question 15

What does it mean for the determinant of a matrix when two rows (or columns) of the matrix are equal?

Answer 15

When two rows (or columns) of a matrix are equal then the determinant of the matrix will be zero.

Note: This is also the case if a row or column is a scalar multiple of another.

Question 16

What does it mean for the determinant of a matrix when two rows (or columns) of the matrix are interchanged?

Answer 16

When two of the rows, or columns, in a matrix are interchanged then the sign of the resultant determinant is the reverse of the original determinant.

(think of the sign array)