Lecture 28: Mathematical Insight and Engineering

Matrices

Matrices are commonly used in engineering computations.A matrix generally has more than one row and more than one column.Scalar multiplication and matrix addition and subtraction are performed element by element.

Matrix Operations

TransposeMultiplicationExponentiationInverseDeterminantsLeft division

Transpose

In mathematics texts you will often see the transpose indicated with superscript T AT

The MATLAB syntax for the transpose is A'

The transpose switches the rows and columns

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

121110

987

654

321

A⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

12963

11852

10741TA

The dot product is sometimes called the scalar productThe sum of the results when you multiply two vectors together, element by element.

Dot Products

*||

*||

*||

+ +

Equivalent statements

Matrix Multiplication

Matrix multiplication results in an array where each element is a dot product. In general, the results are found by taking the dot product of each row in matrix A with each column in Matrix B

A: m x nB: n x p

€

C(i, j) = A(i,k) *B(k, j)k=1

n

∑

Because matrix multiplication is a series of dot products the number of columns in

matrix A must equal the number of rows in matrix B

For an mxn matrix multiplied by an nxp matrix

m x n n x p

These dimensions must match

The resulting matrix will have these dimensions

Matrix Powers

Raising a matrix to a power is equivalent to multiplying itself the requisite number of times A2 is the same as A*A A3 is the same as A*A*A Raising a matrix to a power requires it to have the same number of rows and columns

Matrix Inverse

MATLAB offers two approaches The matrix inverse function

inv(A) Raising a matrix to the -1 power

A-1

A matrix times its inverse is the identity matrix

Equivalent approaches to finding the inverse of a matrix

Not all matrices have an inverse

Called Singular Ill-conditioned matrices

Attempting to take the inverse of a singular matrix results in an error statement

Determinants

Related to the matrix inverseIf the determinant is equal to 0, the matrix does not have an inverseThe MATLAB function to find a determinant is det(A)

|A| = A(1, 1)*A(2, 2)

- A(1, 2)*A(2, 1)

|A| = A(1,1)*A(2,2)*A(3,3)

+ A(1, 2)*A(2,3)*A(3,1)

+ A(1,3)*A(2,1)*a(3,2)

- A(3,1)*A(2,2)*A(1,3)

- A(3,2)*A(2,3)*A(1,1) - A(3,3)*A(2,1)*A(1,2)

Solutions to Systems of Linear Equations

3 2 10

3 2 5

1

x y z

x y z

x y z

+ − =− + + =

− − = −

Using Matrix Nomenclature

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−=

111

231

123

A⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

z

y

x

X⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

1

5

10

B

and

AX=B

We can solve this problem using the matrix inverse approach

This approach is easy to understand, but its not the more efficient computationally

Matrix left division uses Gaussian elimination, which is much more efficient, and less prone to round-off error

Q. What is the output of the following code fragment? int j; for (j = 0 ; j < 4; j++) { printf(“%d ”, j); j++; } A) 0 1 2 3 B) 0 2 C) 1 3 D) 1 2 3

Practice QuestionS

olu

tion

: B

Q. What is the output of the following program? #include <stdio.h> int myFunc(int a, int *b); int main( void ) { int a = 4; int b = 5; myFunc(a, &b); printf ("a + b = %d\n", a + b); } int myFunc(int a, int *b) { a = a + 2; *b = *b - 1;

return a; }

A) a + b = 9 B) a + b = 8 C) a + b = 11 D) a + b = 10

Practice Question

Solu

tion

: B