Topic 7: Matrix and vector algebra

Question 1

What is a matrix (2)

Answer 1

-A matrix is a rectangular array of numbers which are considered an entity
-Each matrix consists of a series of m rows and n columns

Question 2

How can we draw a matrix (2)

Answer 2

-Draw a shape with 2 [ ] brackets
-Fill this shape in with elements
-Each A_ij is called an element of a matrix, where i = row number, j = column

Question 3

What is the dimension/order of a matrix (2)

Answer 3

-The dimension/order of a matrix corresponds to its numbers of rows and columns
-a 2x3 matrix has 2 rows, 3 columns

Question 4

What is the relationship between matrices and vectors (2,1)

Answer 4

-A vector is a matric where m or n = 1
-This is either a column or row vector

-We normally denote matrices with bold upper case letters, and vectors with lower case

Question 5

What is a null and identity matrix (1,2)

Answer 5

-A null matrix is a matrix where a_ij = 0 for all i and j

-An identity matrix is a square matric where a_ij = 0 for all i ≠ j, and a_ij = 1 for all i = j
-This is the equivalent of multiplying by 1

Question 6

What is a square and triangular matrix (2)

Answer 6

-A square matrix is one where n = m
-A triangular matrix is a square matrix that has only zeros either above or below the main diagonal (MD is where i = j)

Question 7

How can we write a system of 3 equations and 3 variables in matrix form (3,3,1)

Answer 7

The 3 equations are:
-a₁₁x₁ + a₁₂x₂ + a₁₃x₃ = b₁
-a₂₁x₁ + a₂₂x₂ + a₂₃x₃ = b₂
-a₃₁x₁ + a₃₂x₂ + a₃₃x₃ = b₃

-Make A a 3x3 matrix, where a₁₁ is row 1 column 1 etc…
-Make x a 3x1 column matrix, with x₁ row 1…
-Make b a 3x1 column matrix, with b₁ row 1…

(could replace 3 and 3 for m and n)

Question 8

How do you add matrices (2)

Answer 8

-You add each individual element
-This requires the same dimension from the matrices

Question 9

How do you compute AB, where A and B are matrices (1,2,1)

Answer 9

-To compute AB, we multiply each row of A with each column of B

-Imagine matrix A, a 2x2 with elements (going TL, TR, BL, BR) a₁₁, a₁₂, a₂₁, a₂₂
-Also imagine matrix B, a 2x2 with elements (going TL, TR, BL, BR) b₁₁, b₁₂, b₂₁, b₂₂

-AB = (going TL, TR, BL, BR) a_{11b11 + a12b21, a11b12 + a12b22, a21b11 + a22b21, a21b12 + a22b22}

Question 10

How do we know how many rows and columns will be in a matrix which came from multiplying 2 others (2,2)

Answer 10

-Suppose A = [a_ij]_mxn, B = [b_ij]_nxp
-AB = C = [c_ij]_mxp

-To multiply 2 matrices, the number of columns in the first matrix = the number of rows in the second
-The resulting matrix will then have the number of rows in the first and number of columns in the second