Test 1 prep Flashcards
3 types of possible solutions:
1 ) Unique solution
2 ) Infinite solution
3 ) No solution
Unique and infinite solutions are ______ solutions
consistent
No solution is _________
inconsistent
Difference between REF and RREF:
In RREF, the entries aside from leading ones have to be zeros whereas in REF, we only need leading ones as the first entry of a row
True or false: a matrix can have multiple REFs, but only a unique RREF.
True
“Number of leading variables of a matrix is called the _______ of that matrix”
rank (represented as IR^n)
In the parametric solutions, there are 2 types of variables:
1 ) Leading variables
2 ) Free variables
Only a square n x n matrix will have a main _______
diagonal
Entries of a matrix are represented as:
a subscript ij
In a subscript ij, what does ij represent:
i = row
j = column
Matrix product can only be achieved if:
nº of columns in A is identical to nº of rows in B
What is the proper way of writing a system if linear equations in matrix form?
AX = b
What is a matrix transpose?
The interchange of rows and columns
What are the 3 properties of AA^T or A^TA?
1 ) products are always square matrices
2 ) products are always symmetrical matrices
3 ) entries of the main diagonal are always positive
What is the trace of a matrix?
Sum of main diagonal entries
tr(A+B) =
tr(A) + tr(B)
True or false: tr(A) = tr(A^T)
True
tr(k*A) =
k*tr(A)
tr(AB) =
tr(BA)
A + B =
B + A
(A + B) + C =
A + (B + C)
In general, AB does not =
BA
(AB)C =
A(BC)
A(B + C) =
AB + AC (right distribution)
(A + B)C =
AC + BC (left distribution)
k(AB) =
(kA)B = A(kB)
(a + b)C =
aC + bC
(ab) C =
b(aC) = a(bC)
What is the identity matrix?
square matrix (size depending on the other matrices in the equation) where all its entries are 0 and the main diagonal entries are 1
A*A^-1 =
I
How do you find determinant of 2x2 matrix?
det(a) or IAI
ad–bc
How do you find inverse of 2x2 matrix?
A^ -1 = 1/det(A) [exchange a with d; exchange signs of b and c]
A^n =
A multiplied n number of times
A^0 =
I subscript matrix size
A^n * A^r =
A^(n+r)
(A^n)^r =
A ^ (n*r)
A^ -n =
(A^-1)^n = (A^n)^-1
(A^-1)^-1 =
(A^T)^T =
A
(A^-1)^n =
(A^T)^-1 =
(A^n)^-1
(A^-1)^T
(A+B)^-1 =
(A+B)^T =
(A^-1 + B^-1)
(A^T + B^T)
(k*A)^-1 =
(k*A)^T =
k^-1 * A^-1 = (1/k) * A^-1
k * A^T
(ABCD)^-1 =
(ABCD)^T =
D^-1 * C^-1 * B^-1 * A^-1
D^T * C^T * B^T * A^T
How do you find A in AX=b?
X=A^-1 * b
What is a homogenous system?
solutions are = 0
What is a diagonal matrix?
All entries beyond the main diagonal are 0 in a square matrix
What are the 2 properties of the diagonal matrix?
1 ) the diagonal matrix is invertible if none of its main diagonal entries is 0
2 ) the power of the matrix can be distributed to the entries of the main diagonal
What does the LOWER triangular matrix look like?
entries above main diagonal are 0
What does the UPPER triangular matrix look like?
entries below main diagonal are 0
If any entry of the main diagonal of the triangular matrix is 0:
it is SINGULAR
SINGULAR =
NOT INVERTIBLE
The main diagonal of a symmetric matrix acts as a _______
mirror line
The transpose of the symmetrical matrix is _______ to original matrix
identical
The inverse, if it exists, of the symmetrical matrix is _______ to original matrix
identical
The inverse of a matrix only exists if:
det(A) does not = 0
