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## Chapter 5.7

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**Chapter 5.7**Properties of Matrices**Basic Definitions**It is necessary to use capital letters to name matrices. Also, subscript notation is often used to name element of a matrix, as in the following Matrix A.**With this notation,**the first row, first column element is a11 (read “a-sub-one-one”); The second row, third column element is a23 and in general the i th row, j thcolumn element is aij**Certain matrices have special names:**because the number of rows is equal to the number of columns. A matrix with just one row is a row matrix, and a matrix with just one column is a column matrix.**Two matrices are equal if they are the same size and if the**corresponding element, position by position are equal. Using this definition, the matrices are not equal (even though they contain the same elements and are the same size), since the corresponding elements differ.**Find the values of the variables for which each statement is**true, if possible.**Find the values of the variables for which each statement is**true, if possible.**Adding Matrices**Addition of matrices is defined as follows. It can be shown that matrix addition satisfies the commutative, associative, closure, identity, and inverse properties.**Special Matrices**A matrix containing only zero elements is called a zero matrix. A zero matrix can be written with any size.**By the additive inverse property, each real number has an**additive inverse: if a is a real number, then there is a real number –a such that a + (-a) = 0 and –a + a = 0**Given matrix A, there is a matrix –A such that**A + -A = 0 The matrix –A has as elements the additive inverses of the elements of A.**Matrix –A is called the additive inverse, or negative, of**matrix A. Every matrix has an additive inverse Subtracting Matrices The real number b is subtracted from the real number a, written a – b, by adding a and the additive inverse of b. That is, a – b = a + (-b)**In practice, the difference of two matrices of the same size**is found by subtracting corresponding elements.**Multiplying Matrices**In work with matrices, a real number is called a scalar to distinguish it from a matix. The product of a scalar K and a matrix X is the matrix kX, each of whose elements is k times the corresponding element of X.**We have seen how to multiply a real number (scalar) and a**matrix. To find the product of two matrices, such as**first locate row 1 of A**and column 1 of B,**Multiply corresponding elements, and find the sum of the**products. • (-3)(-6) + (4) (2) + (2)(3) • + 8 + 6 = 32**The result is the first element for row 1, column 1 in the**product matrix. • (-3)(-6) + (4) (2) + (2)(3) • + 8 + 6 = 32**Now use row 1 of A and column 2 of B to determine the**element in row 1 and column 2 of the product matrix.**Now use row 1 of A and column 2 of B to determine the**element in row 1 and column 2 of the product matrix. (-3)(4) + (4) (3) + (2)(-2) -12 + 12 + -4 = -4**Now use row 2 of A and column 1 of B to determine the**element in row 2 and column 1 of the product matrix.**Now use row 2 of A and column 1 of B to determine the**element in row 2 and column 1 of the product matrix. (5)(-6) + (0) (2) + (4)(3) -30 + 0 + 12 = -18**Now use row 2 of A and column 2 of B to determine the**element in row 2 and column 2 of the product matrix.**Now use row 2 of A and column 2 of B to determine the**element in row 2 and column 2 of the product matrix. (5)(4) + (0) (3) + (4)(-2) 20 + 0 + -8 = 12**same size**Can the product AB be calculated? The following diagram shows that AB can be calculated, because the number of columns of A is equal to the number of rows of B.**same size**If AB can be calculated, how big is it? size of AB = 3 x 4**If BA can be calculated?**BA cannot be calculated? different size