How to find the determinant of a 2x2 matrix
(Top left x Bottom right) - (Bottom left x Top right)
How to find X and Y from the following system of equations (Cramer’s Rule) :
A1X + B1Y = C1
A2X + B2Y = C2
Write matrix Ax by replacing first column (a’s) with the C’s.
Write matrix Ay by replacing second columns (b’s) with the C’s.
X = det Ax / det A
Y = det Ay / det A
What does det A = 0 suggest about the number of solutions
No solutions
Infinite solutions
How many equations and how many unknowns do there have to be to use Cramer’s rule
Same number of linear equations as the number of unknowns.
How to find the determinant of a 3x3 matrix
Expand along row 1 using the sign pattern + − +.
For each number in row 1:
Multiply the number by the determinant of the 2×2 matrix left after blocking row 1 and that number’s column.
Apply the signs + − + from left to right.
Add the three results together to get the determinant.
What is a permutation
A rearrangement of numbers
What is an inversion
When a larger number appears before a smaller number in your permutation
E.g. In the permutation 2,3,1:
(2,3) and (3,1) are inversions
What is a configuration
When in an nxn matrix you choose exactly n elements such that:
No 2 elements are in the same row
No 2 elements are in the same column
How to find the determinant of a 3x3 matrix using configurations
Find every single way of getting a number from each column and row but CANNOT be in the same row or column, then add them all. If there are any odd number of inversions then it gets a (-) sign.
Det A = A11A22A33 + A12A23A31 + A13A21A32 - A13A22A31 - A12A21A33 - A11A23A32
You only look at the permutation of column indices NOT the actual matrix values.
What is the necessary condition for Cramer’s Rule
Det A MUST NOT = 0
The rule doesn’t work if Det A = 0 as you would be dividing by 0
What is transposition
The rows of A become the columns of A^T
What are the properties of determinants
Det A = Det (A^T) for each square matrix A
If you swap the positions of two different rows in a square matrix, then Det A’ = - Det A
If you multiply each row of a square matrix A by c then det A’ = c x Det A
If in a square matrix, two rows are multiples of each other (3rd row is irrelevant) then determinant = 0
What is the necessary condition for matrix addition
Both matrices must have the SAME number of rows and columns
Common rules about matrices
A + B = B + A
(A + B) + C = A + (B + C)
A + 0 = A
A - A = 0
What is the necessary condition for matrix multiplication
Number of rows in A must equal the number of columns in B
Is matrix multiplication commutative
No
How to multiply 3x3 matrices
(R1 x C1) + (R1 x C2) + (R1 x C3) = new row 1
Repeat to find rows 2 and 3 substituting R1 for R2 and R3
What is the identity matrix
An nxn square matrix with 1’s on the diagonal and 0’s elsewhere
What does multiplying by the identity matrix do
Nothing, same as multiplying by 1
When is a square matrix non-singular/ invertible
If there exists another matrix A^-1 such that AA^-1 = identity matrix
What does an invertible/non-singular matrix mean
Determinant is not 0
How to solve a system of linear equations with matrices
Write in the form Ax = b where A is the matrix, x is the column of unknowns and b is the column of answers.
X = A^-1 x b if A is invertible (it will have 1 solution)
How do we find the inverse of a 2x2 matrix
A^-1 = 1 / Det A x A*
Where to get A*, swap the top to bottom diagonal and multiply bottom to top diagonal by -1
How to find the inverse of a 3x3 matrix
A^-1 = 1 / Det A x A*
Find 9 determinants and use +, -, +, - , + etc..
REMEMBER NOT TO MULTIPLY BY THE NUMBER LIKE YOU DO WHEN FINDING DET OF A 3X3. YOU ARE JUST FINDING THE DET OF THE 2X2’s
Block first row and first column, then second, then third
Repeat but blocking second row instead of first
Repeat but block third row
Then transpose the matrix formed (swap rows and columns)
Now this new matrix is A*
