Section 6.1 Matrix solutions to Linear Systems
a. Matrix Row Operations
1. Any two rows may be inter changed at any time.
2. You may multiply a row by any nonzero number.
3. You may add a multiple of a row to another row.
b. To solve a linear system using Gauss-Jordan elimination, you must use row
ope rations to get 1’s down the main diagonal and zeros both above and below the
1’s.
Section 6.2 Inconsistent and Dependent Systems
a. If Gauss-Jordan elimination results in a matrix with a row containing all
zeros
except for the last entry in the row, the system has no solution.
b. If Gauss-Jordan elimination results in a matrix with a row with all zeros,
the
system has an infinite number of solutions(Contains dependent equations)
Section 6.3 Matrix Operations
a. Two matrices are equal if and only if they have the same order and
corresponding
elements are equal.
b. Matrix addition and Subtraction: Matrices of the same order are added or
subtracted by adding or subtracting corresponding elements.
c. Scalar Multiplication: If A is a matrix and c is a scalar, then cA is the
matrix
formed by multiplying each element in A by c.
d. Matrix multiplication: The product of an mxn matrix A and an nxp matrix B is
an
mxp matrix AB. Multiplying each element in the ith row of A by the
corresponding element in the jth column of B and adding the products find the
element in the ith row and jth column of AB. Matrix multiplication is not
commutative: AB does not equal BA.
Section 6.4 Multiplicative Inverse of Matrices; Matrix Equations
a. The multiplicative identity matrix In is an n x n matrix with 1’s down the
main
diagonal and zeros elsewhere.
b. Let A be an n x n square matrix . If there is a square matrix A-1 such that
AA-1 = In
and A-1 A = In, then A-1 is the multiplicative inverse of matrix A.
c. Procedure for finding A-1:
1. Form the augmented matrix [ A | I ], where I is the multiplicative Identity
matrix of the same order as the given matrix A.
2. Perform row operations on the augmented matrix in step 1 to obtain a
matrix of the form [ I | B ]. This is equivalent to using Gauss -Jordan
elimination to change A into the identity matrix.
3. Matrix B is A-1
d. Linear systems can be re presented by matrix equations
in the form
AX = B, in which A is the coefficient matrix and B is the constant matrix. The
solution is X = A-1B.
Section 6.5 De terminants and Cramer ’s Rule
a. To evaluate a Second-Order Determinant:
b. To evaluate a nth order determinant, where n > 2 ,
1. Select a row or column about which to expand .
2. For each element aij in the row or column, multiply by (-1)ij times the
determinant obtained by deleting the ith row and the jth column in the
given array of numbers
3. The value of the determinant is the sum of the products found in step 2.
c. Cramer’s rule for solving linear systems using determinants:
where D, Dx, Dy, and Dz
are determinants
