Matrices Study Notes for Civil and Mechanical Engineering

1. Matrix Algebra

     1.1 Operations in the matrices:

1.1.1 Equality of matrices:

Two matrices A = [a_ij] and B = [b_ij] are said to be equal if and only if:

(i) They are of the same order and

(ii) each element of A is equal to the corresponding element of B.

1.1.2 Addition

• It is only possible to add 2 matrices if they are of the same order (i.e. same number of rows and columns).
• If we add matrices A and B to give a result X, then each element of X is simply the sum of the 2 corresponding elements of A and B,

For example

• Since matrix addition is performed simply by adding the individual elements, clearly you will get the same result whatever order you add the matrices in. In math's we say that the operation is both commutative and associative:

A+B = B+A

(A+B)+C=A+(B+C)

1.1.3 Subtraction

• Subtraction of 2 matrices is analogous with addition:

• for example, using the same A and B as before

1.1.4 Scalar Multiplication

• Scalar multiplication, i.e. multiplying a matrix by a number (eg F) is simply a matter of multiplying each element of the matrix by the number:

For example:

• It should be clear from the above that scalar multiplication is commutative, i.e.

3×A = A×3

and also that scalar multiplication is distributive over addition and subtraction, i.e

3×(A+B)=2×A+3×B

1.1.5 Matrix Multiplication

• We can multiply 2 matrices to give a matrix result:

X=A×B

• It is only possible to multiply A and B if the number of columns of A is equal to the number of rows of B (A and B are then said to be conformable).
• If A is an n by m matrix, and B is an m by p matrix, then X will be a n by p matrix. The definition of X is:

For example, we can multiply a 2 by 3 matrix and a 3 by 2 matrix, resulting in a 2 by 2 matrix:

• We see that if we change the order of the 2 matrices. This time we are multiplying a 3 by 2 matrix with a 2 by 3 matrix, and the result is a 3 by 3 matrix, quite different from the previous result:

• This shows that matrix multiplication is not commutative. In fact, if you exchange the matrices, the multiplication may become invalid due to conformability.
• For example, if A is a 2 by 3 matrix and B is a 3 by 3 matrix, it is possible to for the product AB, but not BA.

Matrix multiplication is, however, associative and distributive as:

A×B≠B×A

(A×B)×C=A×(B×C)

(A+B)×C=A×C+B×C

A(B + C) = AB + AC

A(B – C) = AB – AC

(B – C)A = BA – CA

1.1.6 Transposition

• Transposing a matrix means converting and m by n matrix into an n by m matrix, by “flipping” the rows and columns.

• It is denoted by a superscript T, eg:

As an aside, there is an interesting relationship between transposition and multiplication:

(A×B)^T = B^T×A^T

A = (A^t)^t = A

(A +B)^t = A^t + B^t

(kA)^t = k(A)^t

1.2   Trace of Matrix:

Let A be a square matrix of order n. The Sum of elements lying along the principal diagonal is called the trace of A denoted by Tr(A).

1.2.1 Properties of trace of matrix:

(a). tr (λA) = λ tr(A)

(b). tr (A +B) = tr (A) + tr (B)

(c). tr (AB) = tr (BA)

1.3 Adjoint and Inverse of the Matrix:

1.3.1 Adjoint of a square matrix:

 1.3.2 Inverse of a matrix:

If A be any matrix, then a matrix B if it exists, such that:

 AB = BA = I

Then, B is called the Inverse of A which is denoted by A^-1 so that AA^-1= I.

1.3.2.1 Properties of Inverse

(a). AA^–1 = A^–1 A = I

(b). A and B are are inverse of each other iff AB = BA = I

(c). (AB)^–1 = B^–1 A^–1

(d). (ABC)^–1 = C^–1 B^–1 A^–1

(e). If A be a n × n non-singular matrix, then (A’)^–1 = (A^–1)’.

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