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[MA212/DE2] Formulas to know for 1st exam

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Trivial solution to vector A vector x = vector 0
The trivial solution is vector x=vector 0
Properties of determinants: let the matrix B be a row operation on matrix A where for one of the rows, it is mulitplied by c (c is a constant.) for example, one such operation applied to B is R1 <- cR1. what is det(B)?
c*det(A) (multiply by constant)
Properties of determinants: let matrix B be a row operation on matrix A where for one of the rows, it is added or subtracted to another and one of them has a constant applied to them (c is a constant.) for example, R2 <- 5R1 - R2. what is det(B)?
det(A) (same)
Properties of determinants: let matrix B be a row operation on matrix A where two of the rows are swapped. for example, R2 <-> R1. what is det(B)?
-det(A) (negative)
Let A = [a11 a12
a21 a22 ]
What is the general formula for the determinant of A, by doing the cofactor expansion along A?
(-1)^1+1 * a11*|A11| + (-1)^1+2 * a12 * |A12| (|| denotes the determinant of the cofactor matrices, determined by crossing out the element (for example for |A11| , you cross out a11) using a horizontal and vertical line)
Formula for solving system of equations for x using inverse matrices for the following: A x = B
x = A^-1 B
How to find the inverse of a matrix A using row operations
using the augmented matrix system (A|I), for example [1 2 | 1 0
3 4 | 0 1]->you want to get the left hand side to be an identity matrix. The right hand side is your inverse. YOU WILL NEED TO GO FURTHER THAN ROW ECHELON FORM TO FIND INVERSES.
When in row echelon form, the determinant of a matrix is determined by the products of what?
The main diagonal
Multiply an axb matrix (X) by a cxd matrix (Y): you can only do this when dimensions of the multiplied matrix, XY, are defined. How is this found?
When multiplying XY, you must make sure that b=c
Multiply an axb matrix (X) by a cxd matrix (Y): assuming the multipled matrix XY is defined (b=c), what are the dimensions of XY?
axd
Dimensions of nxn means a
square matrix
Singular (think "a delicate special matrix") means invertible or noninvertible?
noninvertible
Nonsingular (think "a non fragile non special matrix") means invertible or noninvertible?
invertible
You can also see if a matrix is invertible (or nonsingular) by making sure...
det(A)≠0. (it has a nonzero determinant)
When a matrix is invertible, it has ___ solution(s).
exactly 1
When a matrix is invertible, what rank does it have?
Full rank
Full rank definition
???
The sys of eqns given by the augmneted matrix:
[ 1 2 3 | 0
0 9 1 | 0
0 0 0 | 0 ]
has how many solutions?
Infinitely many solutions (more unknowns in row reduced matrix than you can solve for)
The sys of eqns given by the augmneted matrix:
[ 1 2 3 | 0
0 9 1 | 0
0 0 0 | 3 ]
has how many solutions?
No solutions (0x1+...=3)
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