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Definitions Matrix: a set of values stored in a rectangular array. Element: A single variable in the matrix Row: the set of elements on a horizontal line Column: The set of elements on a vertical line Order: The size of a matrix, denoted (row x column). If the matrix is square, the order can be denoted simply as (row). Addition\Subtraction: Add and subtract elements that map to the same row and column in the other matrix to produce a new matrix. Only matrices with identical orders can be added\subtracted. Multiplication by a scalar: Multiply each element by the scalar to produce a new matrix. Example: 1 2 5 4 8 20 48 3 4 32 12 16 7 1 5 28 4 20 Zero Matrix: A matrix with all elements equal to zero. Multiplying another matrix by a zero matrix gives a zero matrix because each element of the non-zero matrix will be multiplied by zero, giving a zero matrix. Identity Matrix: A zero matrix with ones going diagonally from the upper left corner to the bottom right. The importance of an identity matrix is that when it is multiplied by another matrix, it returns that same matrix. In other words, if I is the identity matrix and X is another square matrix, I*X = X Inverse of a Matrix: As long as the determinant is not zero, the inverse of a 2x2 matrix is 1 A 1 d b a b 1 * = = For example, a * d b * c c a c d 1 A 1 7 3 4 3 1 1 7 3 * * = = = 4 * 7 3 * 6 6 4 10 6 4 6 7 The inverse matrix is used in systems of equations to find variables x, y, and z: 1 a b c x j x a b c j d e f y k y d e f k g h i z l z g h i l If the determinant is zero, either there are an infinite number of solutions or there are no solutions. Determinant of a square matrix: Determinants are used to find inverse matrices and to solve linear equations. Absolute value signs around a matrix instead of parentheses symbolize the determinant. A matrix is singular if it has a determinant of zero; otherwise, the matrix is non-singular. Determinant of 2x2 matrix: a b det(A) = |A| = = a *d b*c c d Determinant of a 3x3 matrix: a b c e f d f d e b* c* det(A) = |A| = d e f = a * h i g i g h g h i aei bfg cdh ceg afh bdi Row Reduction with Augmented matrices (Assuming the determinant is not zero): The goal of the row reduction process is to give zeroes in all elements below the diagonal. To do this, we may swap any two rows or add a constant multiple of any row to another row. With combinations of these two steps, all non-zeros below the diagonal. A reduced matrix A is referred to as A Re d . An Augmented matrix is a combination of two matrices to create a matrix that is easier to handle. This is usually done to simplify systems of equations. An example: 3x + 4y + z = 7 2x – y + 4z = 3 5x + 6y +z = 13 4 1 x 7 3 2 1 4 y 3 Making the system into an augmented 5 6 1 z 13 matrix by removing the (xyz) matrix and The equals sign to combine the two number matrices with a vertical line to separate them: 3 4 1 7 2 1 4 3 Subtract 2nd row from 1st 5 6 1 13 1 5 3 4 2 1 4 3 5 6 1 13 1 5 3 4 Subtract 2 times the 1 from the 2 0 11 10 5 Add 5 times the 1st to the 5 6 1 13 1 5 1 5 3 4 3 4 3rd 0 11 10 5 Add 31 (*) the 2nd to 11 (*) the 3rd 0 11 10 5 0 31 14 33 0 0 464 518 st nd To complete a row reduction, the second row should be divided by -11 and the last row should be divided by 464 to give ones along the diagonal. However, before doing so, it 259 can be concluded that 464z = 518 or z = . From the 2nd row: 232 259 1875 -11y – 10( ) = 5 --> y = . Lastly, the 1st row gives: 1276 232 1875 259 x + 5( ) – 3( ) = 4 --> x = 14.696. 1276 232