LA Lecture 5 Flashcards
What is the determinant of a 2x2 matrix
A =
a b
c d?
ad - bc
What is the determinant of a 3x3 matrix
A =
a b c
d e f
g h i
(aei + bfg + cdh) - (gec +hfa + idb)
What is the determinant of a singular matrix?
0
What happens to the determinant if two rows of a matrix are exchanged?
The determinant changes sign (sign reversal)
What is the determinant of a matrix after elimination (EA)?
The determinant is the product of the pivots, which equals the determinant of the original matrix (det EA = det A).
a ( d - [c/a]b)
What is the determinant of the n by n identity matrix?
1
What happens to the determinant if a row is multiplied by a number
t?
The determinant is multiplied by t
What is the determinant if two rows of a matrix are equal?
The determinant is 0.
What happens to the determinant if a multiple of one row is subtracted from another row?
The determinant remains unchanged.
What is the determinant of a matrix with a row of zeros?
0
What is the determinant of a triangular matrix?
The product of the diagonal entries.
What is the determinant of a diagonal matrix?
The product of the diagonal entries.
What is the determinant of a singular matrix?
0
What is the determinant of the product of two matrices AB?
det A x det B
What is the determinant of the transpose of a matrix A^T?
The same as the determinant of A
( det A^T = det A)
What is the determinant of a 2x2 matrix
A =
a b
c d
ad - bc
How many terms are in the determinant of an n×n matrix using the Big Formula?
n! terms
What is the cofactor Cij of a matrix A?
Cij = (-1)^i+j det Mij,
What is the relationship between the determinant of a matrix and its pivots in LU decomposition?
det A = d1d2…dn
What is the determinant of a triangular matrix
The product of the diagonal entries.
What is the determinant of the identity matrix?
1
What happens to the determinant if two rows of a matrix are exchanged?
The determinant changes sign (sign reversal).
What is the formula for the inverse of a matrix A using cofactors?
A^-1 = C^T/ det A
where C is the cofactor matrix.
What is the formula for xij in Cramer’s Rule?
x j = det Bj / det A
What is the area of a parallelogram formed by vectors (a,b) and (c,d)?
the area is |ad-bc|
What is the volume of a box (parallelepiped) formed by the columns of a matrix A?
the volume is |det A|
What is the cross product of two vectors u and v?
The cross product u×v is a vector perpendicular to both u and v
What is the relationship between v×u and u×v?
v x u = - (u x v)
what is the area of a triangle?
area = 1/2 determinant
What is the cross product of i=(1,0,0) and j=(0,1,0)?
i x j = k, where k = (0,0,1)
What is the determinant of a matrix A if its rows (or columns) are the sides of a box?
The determinant detA gives the volume of the box.
What is the volume of a box formed by the columns of a 3×3 matrix A?
volume is |det A|
cross product of u = (1,1,1) and v = (1,1,2)
(1, -1, 0)
What is the determinant of a matrix A if its rows are linearly dependent?
If the rows are linearly dependent, det A = 0
