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 (3)

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) a₁₁b₁₁ + 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

Question 11

What are some rules with matrix multiplication (4)

Answer 11

-AB ≠ BA
-If AB = 0, this doesn’t necessarily mean A, B or BA = 0
-AB = AC doesn’t; necessarily mean B = C
-AB means B is premultiplied by A or A is postmultiplied by B

Question 12

What is an idempotent matrix (1)

Answer 12

-A square matrix A is said to be an idempotent matrix is A² = A

Question 13

What is the transpose of a matrix + example (3)

Answer 13

Let A be an (mxn) matrix
-The tranpose of A corresponds to the (nxm) matrix, whose first column is the first row of A, etc
-This is denoted A’ or A^T
-If A = (TL TR BL BR) = [1, 2, -1, -2], A’ = [1, -1, 2, -2]

Question 14

When is a matrix symmetric (1)

Answer 14

-If it is equal to its transpose

Question 15

How can we use matrices to solve a system of equations (2)

Answer 15

-Take Ax = b, for a square matrix A_nxn, X_nxm, b_nxm
-If the inverse of A exists, we can premultiply the equation by A^-1 and get x = A^-1b

Question 16

How do we work out the determinant of a 2x2 matrix (1)

Answer 16

-Let A = (TL TR BL BR) [a, b, c, d], Det |A| or |A| = ad-bc

Question 17

What is the determinant of the 2x2 hessian matrix (2)

Answer 17

-Remember the Hessian Matrix as (TL TR BL BR) [f₁₁’’, f₁₂’’, f₂₁’’, f₂₂’’]
-|H| = f₁₁'’f₂₂’’ - (f₂₁’’)²

Question 18

How do we work out the determinant of matrices where n≥3, and what is the formula for a 3x3 determinant (2,3)

Answer 18

-When n ≥ 3, we try to break down the determinant of the big matrix into the determinants of smaller matrices
-The determinants of these smaller matrices are called minors

-Consider 3x3 matrix (From TL to TR, ML to MR, BL to BR) [a₁₁ a₁₂ a₁₃, a₂₁ a₂₂ a₂₃, a₃₁ a₃₂ a₃₃
]
-|A| = a₁₁|a₂₂a₂₃, a₃₂a₃₃| - a₁₂|a₂₁a₂₃, a₃₁a₃₃| + a₁₃|a₂₁a₂₂, a₃₁a₃₂|
-Multiply all the elements in one row by the determinants of the 2x2 matrix given if you eliminate that row and column, then remember to alternate between + (TL), - (TM) etc

Question 19

What is the formula of the determinant to the (nxn) matrix + what is the cofactor (1,2)

Answer 19

• |A| = a_i1C_i1 + a_i2C_i2 + … + a_ijC_ij

-C_ij corresponds to the cofactor of the element a_ij, where you have to delete the i^th and j^th column.
-Thus, the cofactor C_ij corresponds of the minor M_ij (determinant of smaller matrix) multiplied by (-1)^i+j

Question 20

What are some rules about the determinant of a matrix (7)

Answer 20

• |A’| = |A|
  -If all the elements in a row/column = 0, |A| = 0
  -If 2 rows/columns are interchanged, the determinant changes sign but its absolute value remains the same
  -If all the elements in a single row/column are multiplied by a, |A| = a|A|
• |bA| = bⁿ|A|
• |AB| = |A||B|
• |A+B| isn’t equal to |A| + |B|

Question 21

What is an inverse matrix (2)

Answer 21

-Let A be an (nxn) matrix
-The inverse matrix A^-1 of A is the matrix that solves A(A^-1) = A^-1(A) = Identity matrix

Question 22

What is the formula for a 2x2 inverse matrix (2)

Answer 22

-Let A = (TL TR BL BR) [a, b, c, d]
-A^-1 = (1/ad-bc)[d, -b, -c, a]

Question 23

What is the formula for the inverse of a (nxn matrix (1,3)

Answer 23

-Theorem: any square matrix A_nxn with determinant |A| ≠ 0 has a unique inverse A^-1

-A^-1 = (1/|A|)adj(A)
-adj(A) is the adjoint of A corresponding to (C⁺)’
-(C⁺)’ is the matrix of cofactors, transposed

Question 24

How to work out the inverse of a matrix (4)

Answer 24

-Work out the determinant
-Work out the cofactors (get rid of row and column, work out determinant, do + or -)
-Transpose the matrix of the cofactors (Swap rows and columns)
-1/det x this transposed matrix

Question 25

What does it mean if a matrix is nonsingular, and what is a necessary and sufficient condition for this (2)

Answer 25

-A matrix that has an inverse is non singular -A necessary and sufficient condition for a matrix A_nxn to be nonsingular is |A| ≠ 0

Question 26

What is the length of a vector (2)

Answer 26

-The length of a vector is the distance between the origin and the point -√a₁² + ... + a_n²

Question 27

When are 2 vectors said to be orthogonal (2)

Answer 27

-If ab = 0 -If they are perpendicular