If m and n are positive integers, then an m n matrix is a rectangular array in which each entry a ij of the matrix is a number. The matrix has m rows and n columns.
A real matrix is a matrix all of whose entries are real numbers. i (j) is called the row (column) subscript. An m n matrix is said to be of size (or dimension ) m n. If m=n the matrix is square of order n. The a i,i ’s are the diagonal entries.
Given a system of equations we can talk about its coefficient matrix and its augmented matrix. These are really just shorthand ways of expressing the information in the system. To solve the system we can now use row operations instead of equation operations to put the augmented matrix in row echelon form.
1. Interchange two rows. 2. Multiply a row by a nonzero constant. 3. Add a multiple of a row to another row.
Two matrices are said to be row equivalent if one can be obtained from the other using elementary row operations. A matrix is in row-echelon form if: › All rows consisting entirely of zeros are at the bottom. › In each row that is not all zeros the first entry is a 1. › In two successive nonzero rows, the leading 1 in the higher row is further left than the leading 1 in the lower row.
1. Write the augmented matrix of the system. 2. Use elementary row operations to find a row equivalent matrix in row-echelon form. 3. Write the system of equations corresponding to the matrix in row- echelon form. 4. Use back-substitution to find the solutions to this system.
In Gauss-Jordan elimination, we continue the reduction of the augmented matrix until we get a row equivalent matrix in reduced row- echelon form. (r-e form where every column with a leading 1 has rest zeros)
A system of linear equations in which all of the constant terms is zero is called homogeneous. All homogeneous systems have the solutions where all variables are set to zero. This is called the trivial solution.
An n by n matrix is called an elementary matrix if it can be obtained from I n by a single elementary row operation. These matrices allow us to do row operations with matrix multiplication.
Theorem: Let E be the elementary matrix obtained by performing an elementary row operation on I n. If that same row operation is performed on an m by n matrix A, then the resulting matrix is given by the product EA.
These correspond to the three types of EROs that we can do: › Interchanging rows of I -> Type I EM › Multiplying a row of I by a constant -> Type II EM › Adding a multiple of one row to another -> Type III EM
E1 = How is this created? Eg. 1 Suppose A = E1A = = What is AE1?
Let A and B be m by n matrices. Matrix B is row equivalent to A if there exists a finite number of elementary matrices E 1, E 2,... E k such that B = E k E k-1... E 2 E 1 A.
This means that B is row equivalent to A if B can be obtained from A through a series of finite row operations. If we then take two augmented matrices (A|b) and (B|c) and they are row equivalent, then Ax = b and Bx=c must be equivalent series
If A is row equivalent to B, B is row equivalent to A If A is row equivalent to B and B is row equivalent to C then A is row equivalent to C
Require square matrices Each square matrix has a determinant written as det(A) or |A| Determinants will be used to: › characterize on-singular matrices › express solutions to non-singular systems › calculate dimension of subspaces
If A and B are square then It is not difficult to appreciate that If A has a row (or column) of zeros then If A has two identical rows (or columns) then
If B is obtained from A by ERO, interchanging two rows (or columns) then If B is obtained from A by ERO where row (or column) of A were multiplied by a scalar k, then
If B is obtained from A by ERO where a multiple of a row (or column) of A were added to another row (or column) of A then
That Is the determinant is equal to the product of the elements along the diagonal minus the product of the elements along the off-diagonal.
Note: The matrix A is said to be invertible or non-singular if det(A)≠ 0. If det(A) = 0, then A is singular.
Using row 2 - expansion we fix row 2 and find the minors for each entry in row 2 then apply the sign corresponding to each entries position to find the cofactors. The cofactors are then multiplied by the corresponding entry and summed.
If A is square and is in Echelon form then is the product of the entries on the (main) diagonal.
Using CRAMER’S RULE we can apply this method to finding the solution to a system of linear systems that have the same number of variables as equations There are two cases to consider
Consider the square system AX = B where A is n x n. If then the system has either I) No solution or ii) Many solutions If A 1 is formed from A by replacing column 1 of A with column B and I., then the system has NO solution II. |A| ≠ 0, then the system has a unique solution
Consider the square system AX = B where A is n x n. If |A|≠ 0 then the system has a unique solution The unique solution is obtained by using the Cramer’s rule.
Where A i is found from A by replacing column i of A with B.
Use Cramer’s rule to write down the solution to the system