ECE 3301 General Electrical Engineering Presentation 13 Matrices and Determinants

Sets of Equations Electrical networks may be modeled by a set of simultaneous equations of the form:

Sets of Equations The coefficients a11, a12, … , ann and b1, b2, … , bn are known values. They may be complex or real numbers.

Sets of Equations The unknowns to be determined are x1, x2, … , xn.

Sets of Equations This set of equations may be expressed in matrix form:

Sets of Equations Here [A] is an n  n square matrix of known values. They may be real or complex.

Sets of Equations x] is an n-element vector of unknowns.

Sets of Equations b] is an n-element vector of known values.

Sets of Equations The set of equations may be verified by the definition of matrix multiplication.

Sets of Equations The set of equations may be verified by the definition of matrix multiplication.

Sets of Equations The set of equations may be verified by the definition of matrix multiplication.

Sets of Equations The set of equations may be verified by the definition of matrix multiplication.

Sets of Equations The set of equations may be verified by the definition of matrix multiplication.

Sets of Equations This set of simultaneous equations may be solved by a number of mathematical techniques.

Sets of Equations These techniques include Gaussian elimination or inversion of the square matrix [A].

Sets of Equations Solving such sets of equations is more conveniently done numerically using computer programs written for that purpose.

Sets of Equations This method is not computationally efficient, however it is useful for finding general solution for small sets of equations.

Sets of Equations The application of Cramer’s Rule requires the use of determinants.

Determinants of Square Matrices The n  n square matrix [A], has a determinant D, that is denoted:

Determinants of Square Matrices The value of the determinant is found by expanding the determinant in cofactors down any column or along any row. (Pick the 1st column.) Dk1 is the determinant of the square matrix with the kth row and 1st column deleted.

Determinants of Square Matrices Each of the remaining determinants must be expanded in turn. Obviously this is a very tedious process for matrices of significant dimensions. Dk1 is the determinant of the square matrix with the kth row and 1st column deleted.

Determinants of Square Matrices Starting with 2  2 matrices, the process is relatively simple. - +

Determinants of Square Matrices For a 3  3 matrix:

Determinants of Square Matrices The remaining 2  2 matrices are expanded:

Determinants of Square Matrices The products are expanded to reveal:

Determinants of Square Matrices The terms of the equation may be grouped as follows:

Determinants of Square Matrices This procedure may be carried out by repeating the first two columns and multiplying the diagonals as follows: - - - + + +

Properties of the Determinant Property 1: If each element of a column (or row) of a matrix is multiplied by a constant, k, the value of the determinant is multiplied by k.

Properties of the Determinant Property 2: If to any column (or row) of a matrix, there is added k times the corresponding element of any other column (row), the value of the determinant is unchanged.

Properties of the Determinant Property 2: If to any column (or row) of a matrix, there is added k times the corresponding element of any other column (row), the value of the determinant is unchanged.

Cramer’s Rule Consider the set of simultaneous equations:

Cramer’s Rule which may be expressed in matrix form :

Cramer’s Rule The determinant of the n  n square matrix [A] is:

Cramer’s Rule From Property 1 of the determinant we write:

Cramer’s Rule From Property 2 of the determinant we write

Cramer’s Rule This procedure may be continued to write:

Cramer’s Rule Comparing this determinant to the set of equations, we note that the entries in the first column are defined by b1, b2 … bn, respectively.

Cramer’s Rule Making this substitution we may write:

Cramer’s Rule We may simply solve for x1 (provided D  0 ):

Cramer’s Rule The determinant in the numerator is the determinant of the square matrix [A] with the first column replaced by the vector b]

Cramer’s Rule We define this determinant:

Cramer’s Rule This same procedure may be repeated for the remaining columns in the square matrix [A]. Defining Dk as the determinant with the kth column replaced by the vector b]. The solution to the set of equations is:

Cramer’s Rule For a set of three equations:

Cramer’s Rule Written in Matrix Form:

Cramer’s Rule Here [A] and b] are the known values and x] are the unknowns:

Cramer’s Rule The determinants of this system are:

Cramer’s Rule The solution of the system is: