Y1, C6 - Matrices Flashcards
How many rows and columns does a 3 x 2 matrix have
3 rows
2 columns
How to add or subtract matrices
Add / subtract the corresponding matrix
What is a zero matrix
All elements are 0
What is an identity matrix
Square matrix 0s filled with 1s in the leading diagonal
What are the only matrices that can be raised to a power
A
Square matrices
When can you multiply two matrices
When the dimensions agree
The columns of matrix A must equal the rows of matrix B e.g.
(3 x 4) * (4 x 5)
Are matrices commutative or noncommutative and what does this mean
Noncommutative
The order in which you multiply matrices changes the output
What happens when you multiply by an identity matrix (A * I) OR (I * A)
The result will equal A
Matrix multiplication is associative, what does this mean
A(BC) = (AB)C
it doesn’t matter what order you multiply in
What does the determinant of a matrix tell you
If it has an inverse
How do you know if a matrix has an inverse
If the determinant is NOT 0
What is a singular matrix
A matrix with no inverse, determinant = 0
What is a non-singular matrix
A matrix with an inverse, determinant greater or less than 0
Formula for the determinant of a 2 x 2 matrix
(ad) - (bc)
Formula for determinant of a 3 x 3 matrix
a(ei - fh) - b(di - fg) + c(dh - eg)
minor of A - minor of B + minor of C
What is the minor of a 3 x 3 matrix
The determinant of the 2 x 2 matrix left over after the rows and columns of the element you are working out for the minor for have been crossed out
What is M x M^-1
I
What does it mean if a matrix is self inverse
M = M^-1
M * M = I
M^2 = I
How do you inverse a 2 x 2 matrix
A^-1 = 1 / (det(A)) (d , -b, -c, a)
(1 / determinant of A) * matrix with a and b swapped, b and c negated)
BAB = I, prove A = B^-1 * B^-1
BAB = I
B^-1 BAB = B^-1 I
IAB = B^-1
AB = B^-1
A = B^-1 B^-1
Prove that (PQ)^-1 = Q^-1 * P^-1
(PQ)(PQ)^-1 = I
Q(PQ)^-1 = P^-1
(PQ)^-1 = Q-1 * P^-1
What is the transpose of a matrix (A^T)
The rows and columns are interchanged
m x n becomes n x m
What is a matrix of cofactors (C)
A matrix which follows the pattern of +, -, +, -
How to find the inverse of a 3 x 3 matrix
1) Find det(A)
2) Form a matrix of minors
3) Form a matrix of cofactors (negate every other element)
4) A^-1 = (1 / det(A)) * C^T
Inverse = (1 / determinant) * Transpose of matrix of cofactors
How can you use a matrix to solve simultaneous equations
If A(x, y, z) = v then (x, y, z) = A^-1 * v
What does it mean if a system of equations is inconsistent
There are no solutions
What is a sheaf
When 3 planes intersect along a line
What is a prism in matrices
When planes intersect in pairs but don’t all intersect at any one point
What does it mean if a system of equations is consistent
There is at least one solution of all three equations
How would you know if 3 simultaneous equations are consistent or inconsistent, if there have one, zero, or infinite solutions and if they: meet at a point, a plane, a sheaf, parallel, or create a prism
No solutions, parallel or prism, use 2 of the equations to eliminate a variable and if remaining equations are multiples it is parallel
If infinite solutions, sheaf or same planes, if all planes are multiples they are a plane
One solution, all planes meet at one point
