1. EG1C2 Matrix Algebra - Information Sheet
Note, information such as that on this sheet is provided for students when they do their tests and in their Part 1 exam.
Matrix Definitions
aij is element in row i column j of A
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EG1C2 Engineering Maths: Matrix Algebra Data Sheet

2. EG1C2 Engineering Maths: Matrix Algebra Data Sheet
Determinants
Determinant of A, detA = |A| =
|a11| = a11
Determinant of a matrix A omitting ith row and jth column is Mij
The cofactor of element i,j, Cij, is (-1)i+jMij
|A| = a11*C a12*C a1n*C1n
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EG1C2 Engineering Maths: Matrix Algebra Data Sheet

3. Adjoint, Inverse and Transpose
If A is square, the Adjoint matrix of A, Adj(A), = [Cji] = [Cij]T
(A-1)-1 = A A * A-1 = A-1 * A = I (A * B)-1 = B-1 * A-1
AT is the transpose of matrix A.
(AT)T=A (A+B)T=AT+BT (A*B)T=BT*AT
A is a symmetrix matrix if AT=A.
A is a skew-symmetrix matrix, if AT=-A
If AT=A-1 then A is an orthogonal matrix.
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EG1C2 Engineering Maths: Matrix Algebra Data Sheet

4. Gaussian Elimination: to solve for x in A x = b
Then process until A part is triangular, using row operations
row X := p * row Y + q * row Z
p4 RJM 08/08/02
EG1C2 Engineering Maths: Matrix Algebra Data Sheet

5. EG1C2 Engineering Maths: Matrix Algebra Data Sheet
Gauss Jordan
To invert matrix A
Then do row operations until left half = I, A-1 is right half
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EG1C2 Engineering Maths: Matrix Algebra Data Sheet

6. EG1C2 Engineering Maths: Matrix Algebra Data Sheet
Cramer’s Theorem
A linear system, described by equation A x = b, has solutions:
x1 = D1/D
x2 = D2/D
... xm = Dm/D
where D is |A|
and Dk is determinant of A when kth column replaced by b.
Rank of m*n matrix
Is the largest square submatrix whose determinant is non zero. A submatrix is a matrix with any row and/or column removed.
p6 RJM 08/08/02
EG1C2 Engineering Maths: Matrix Algebra Data Sheet

7. Transformation Matrices
For transforming point x,y to point x’,y’
If T is product of such matrices, x,y transformed to x’,y’ by
p7 RJM 08/08/02
EG1C2 Engineering Maths: Matrix Algebra Data Sheet

8. Eigenvalues and Eigenvectors
Suppose A is an n*n matrix, l a scalar and x an n element vector.
An eigenvalue of A is any value for l which is a solution to the equation A*x - l*x = 0 for which x <> 0.
The n eigenvalues are found by solving |A - l*I| = 0.
The n eigenvectors are the n vectors which satisfy
(A - l*I) x = 0
for each of the n values of l.
State Space Equations
p8 RJM 08/08/02
EG1C2 Engineering Maths: Matrix Algebra Data Sheet

9. State Space Solutions, for A is 2*2
If A has 2 real non repeating eigenvalues l1 and l2 and eigenvalues L1 and L2, general solution is
x = c1*L1*el1t + c2*L2*el2t (c1 and c2 are constants).
If eigenvalues are complex, l1, l2 = a  jb, and eigenvectors are L1, L2 = A  jB, then the response is (for constants ca and cb):
x = eat { ca A (cos (bt) + sin (bt) ) + cb B (cos (bt) – sin(bt) ) }
{Equivalent to the above expression: ca=c1+c2 and cb=j(c1-c2)}
If A has repeated eigenvalue l and eigenvector  the response is
x = c1*L*elt + c2*(t L+V) *elt
where V is the solution of (A - lI)V = 
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EG1C2 Engineering Maths: Matrix Algebra Data Sheet

10. EG1C2 Engineering Maths: Matrix Algebra Data Sheet
Two Port Networks
Can be defined by:
‘A’ matrices for the following simple elements:
If two elements in series are described by matrices A1 and A2, the matrix describing the elements combined is A1 * A2
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EG1C2 Engineering Maths: Matrix Algebra Data Sheet

11. EG1C2 Engineering Maths: Matrix Algebra Data Sheet
These networks can also be defined by the ‘Z’, ‘Y’ or ‘H’ matrices
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EG1C2 Engineering Maths: Matrix Algebra Data Sheet

12. EG1C2 Engineering Maths: Matrix Algebra Data Sheet
Table to convert between these two port network matrices
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EG1C2 Engineering Maths: Matrix Algebra Data Sheet