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Matrices. A matrix, A, is a rectangular collection of numbers. A matrix with “m” rows and “n” columns is said to have order m x n. Each entry, or element,

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Presentation on theme: "Matrices. A matrix, A, is a rectangular collection of numbers. A matrix with “m” rows and “n” columns is said to have order m x n. Each entry, or element,"— Presentation transcript:

1 Matrices

2 A matrix, A, is a rectangular collection of numbers. A matrix with “m” rows and “n” columns is said to have order m x n. Each entry, or element, in a matrix is denoted by a ij, where i stand for the row number and j stands for the column number.

3 This is an example of a square matrix, because the order is n x n This is an example of a column matrix because it is of the form m x 1 This is an example of a row matrix because it is of the form 1 x n Two matrices are equal if and only if both matrices are of the same order and all the corresponding entries are equal

4 Adding & Subtracting

5 Scalar Multiple

6 The transpose of a matrix A, denoted by A’, is formed by swapping the rows for the columns in the matrix

7 Note that a symmetric matrix is symmetrical about the leading diagonal Note that a skew - symmetric matrix must be a square matrix where all entries in the leading diagonal are equal to zero

8 Multiplication of Matrices The matrix product AB can only be found, if the number of columns in matrix A is equal to the number of rows in matrix B This product is possible since there are three columns in A and three rows in B This product is not possible since there are two columns in A and three rows in B

9 Notice that the matrix products of AB is not equal to BA. This is true in general, but not always. If AB should be equal to BA, the matrices are said to be commute.

10 Note that BA cannot be found Note, again, that BA cannot be found

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13 equate entries

14 Note that this method can be extended to find expressions for A 4,A 5,….

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18 This is referred to as the determinant of the matrix A This is referred to as the adjugate or adjoint of the matrix A

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22 Before finding the inverse of a 3 x 3 matrix, we should always evaluate the determinant first to make sure the matrix is invertible. To find the inverse, we use EROs, a method introduced in Unit 1

23 The target is to transform the first matrix into an identity matrix, but to apply all operations to the second matrix as well

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