Matrices and Determinants. Matrices. A matrix is a rectangular arrangement of numbers in rows and columns. Rows run horizontally and columns run vertically. The dimensions of a matrix are stated “ m x n ” where ‘ m ’ is the number of rows and ‘ n ’ is the number of columns.
The result, 11, goes in row 1, column 1 of the answer. Repeat with row 1, column 2; row 1 column 3; row 2, column 1; ...
3 x 2
2 x 3
3 x 3
3 x 2
2 x 1
3 x 1
Imagine crossing out the first row.
And the first column.
Now take the double-crossed element. . .
And multiply it by the determinant of the remaining 2x2 matrix
Now keep the first row crossed.
Cross out the second column.
Finally, cross out first row and last column.
ad - bc
Minor of a12 = 2 is determinant of the 2 x 2 matrix obtained by deleting the 1st row and 2nd column
Determinant of A given by
Determinant of A = -15
If all the elements of any row (or column) of a determinant are multiplied by a quantity (K), the value of the determinant is multiplied by the same quantity.
Given an equation system Ax=d where A is n x n.
|A1| is a new determinant were we replace the first column of |A| by the column vector d but keep all the other columns intact
The expansion of the |A1| by its first column (the d column) will yield the expression
because the elements dinow take the place of elements aij.
This is the statement of Cramers’Rule
Find the solution of
Find the solution of the equation system:
♫ Work this out!!!!
Note that |A| ≠ 0 is necessary condition for the application of Cramer’s Rule. Cramer’s rule is based upon the concept of the inverse matrix, even though in practice it bypasses the process of matrix inversion.
The number ‘r’ is called the rank of the matrix A if
The rank of a matrix A is denoted by p(A).