Matrices Flashcards
Size of Matrix
m rows, n columns
m * n
Compatability for Matrix Multiplication + Resulting Size
Columns in A = Rows in B
If A is m * n, B is m * r
Then matrix is n * r
Echelon Form vs Reduced Echelon Form
Echelon Form:
- 0 Rows are at the bottom of the matrix
- A pivot is to the right of the pivot above it
Reduced Form:
- Pivot is equal to 1
- Pivot is the only non-zero entry in the column
Matrix Rank
Number of non-zero rows in the row echelon form
(also the number of the pivots)
Leading Variables
Variables corresponding to the columns containing pivots
Free Variables
Variables that do not correspond to pivots - they can take arbitrary values
Calculating Free & Leading Variables
n = number of variables
r = leading variables
n-r = number of free variables
What if number of variables = leading variables?
No free variables - either no solutions or a unique solution
3 Possibilities when solving linear systems
- No Solution - happens when the last non-zero row is [0, 0, 0] equaling a non-zero value - system is inconsistent
- Infinite Solutions - when the equations are consistent, and there is at least 1 free variable
- One Solution - when the equations are consistent and there are no free variables
What is Transposing
Swapping Row x of Matrix A when Column X of Matrix A
A(j,i) = At(i,j)
Solving for Det
If 2x2 - use formula sheet
If greater - you have to reduce the matrix
So pick a column of cells, and when you select it - remove the column and the row for that cell
- basically reduce it down to a 2x2 matrix- then you can use the
formula sheet
Determinant Test
A n*n matrix is invertible if and only if det(A) != 0
Cramers Rule
If a linear system Ax=b has a unique solution
Then x = (A^-1)*b
Trig Values - sin cos trig & small medium large
Small: sqrt1/2
Medium sqrt2/2
Large: sqrt3/2
Cos: x
Sin: y
Tan: sin/cos
Trig Identity
Sin^2(theta) + cos^2(theta) =1
