1 Systems of Linear Equations Flashcards
In a system of equations when are all equations satisfied?
At the equilibrium
When does the substitution method become hard?
When there are more equations
When is a system of equations consistent?
When the system has atleast one solution
When is a system of equations inconsistent
When there are no solutions
How should you rearrange equations so Gaussian elimination can be used?
With all the unknowns on the left and parameters on the right
What is a matrix
A rectangular array of elements
Why does an m x n matrix look like?
M rows and N columns
What is a vector?
A matrix with only one row or one column
What notation is used for a transposed matrix
A’ or A ^t if A Is the original matrix
What does transposing a matrix do?
• if matrix A is m x n then matrix A’ is n x m
What is a square matrix
A matrix with the same number of rows and columns
What are symmetric matrices
A matrix where A=A’ this can only happen if the matrix is square
What do you do when multiplying by a scalar?
Multiply all elements by the scalar
When can matrices be added?
When they are the same size
When can matrices be subtracted?
When they are the same size
When can two matrices be multiplied?
When the number of columns of the first is equal to the number or rows of the second
If a matrix (m x n) and a matrix (n x k) are multiplied, what are the dimensions of the result
m x k
Does ABC=BAC if all letters are matrices
No
Does (AB)C=A(BC) if all letters are matrices?
Yes
Does A(B+C)= AB+AC if all letters are matrices?
Yes
Does (AB)’=B’A’
Yes
What is the identity matrix?
A square matrix made up of n rows and columns, where all elements are zeros except for the diagonal which are 1’s
What happens when a matrix of the right dimensions is multiplied by the identity matrix?
It equals the original matrix
A x In = In x A = A
What is the inverse matrix
The inverse matrix of an n x n matrix A is an n x n matrix A^-1 such that A x A^-1 = A^-1 x A = In
When are inverse matrices defined?
For square matrices when the determinant doesn’t equal zero
When is a matrix called singular
When the matrix doesn’t have an inverse
When is a matrix non singular?
When the matrix has an inverse
When are the inverses of matrices unique?
Always
How is the inverse of a 2x2 matrix found
A^-1= 1/|A|x (a22 -a12)
(-a21 a11)
What does (A^-1)^-1 equal?
A
What is (A^-1)’ equal to?
(A’)^-1
What is (AB)^-1 equal to? Assuming AB is invertible
B^-1A^-1
What is (cA)^-1 equal to? Assuming c is a number not equal to 0
c^-1A^-1
What form can a system of linear equations be put into
Matrices in the form Ax= b
What is the determinant used for?
To find the inverse matrix or determine if it exists
When do determinants occur?
Only with square numbers
What is the determinant of a 1x1 matrix?
The only element in the matrix
What is the determinant of a 2x2 matrix
a11a22-a12a21
What is the sarrus rule
A way of finding the determinant in a 3x3 matrix
|A| = a11a22a33 + a12a23a31 + a13a21a32 - a31a22a13 - a32a23a11 - a33a21a12
Diagonal pattern
How can the determinant be found for larger matrices?
They can be broken down until the determinant is easier to find with the below equation
|A|= a11|A11| - a12|A12| + … + (-1)^(1+n)a1n|A1n|
What is the minor of an element
|Aij| with deleted row i and column j is called the minor of element aij
What is the cofactor
The cofactor is the minor with the appropriate sign
Cij = (-1)^(i+j) |Aij|
What is |A’| equal to?
|A|
What is |AB| equal to
|A| x |B|
What is C’ called?
The adjoint of the given matrix, it is the transposed matrix of the cofactors
What are the two ways the inverse can be found?
A^-1 = 1/|A| x C’
A^-1 = 1/|A| x (-1)^(i+j) x |Aji|
What is the way of finding the inverse of a particular element in a matrix
Aij^-1= 1/|A| x Cji
What can be said about the equation system Ax=b if |A| is not equal to zero
There is a unique inverse A^-1
The solution x= A^-1 x b is the only solution
What is cramers rule?
A way of finding the solution to an equation system
X1= |D1|/|A|
Xn = |D1|/|A|
What is the Leontief model?
An input output model, it is a classical application of linear Algebra in economics
Equation for the Leontief model
Xi = ai1x1 + … + ainxn + bi
- xi= total number of good i produced
- aij= units of good i needed to produce one unit of j
- aijxj = number of units of good i needed to produce xj units of good j
- bi = consumption of good i
