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Matrix Properties

Matrix Properties. Some Special Matrices. A Zero matrix is a matrix whose elements are all equal to 0. For instance, here is the 2 x 3 zero matrix: A Square matrix is a matrix that has the same number of rows as columns. For instance, here is a 3 x 3 square matrix.

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Matrix Properties

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  1. Matrix Properties

  2. Some Special Matrices • A Zeromatrix is a matrix whose elements are all equal to 0. • For instance, here is the 2 x 3 zero matrix: • A Square matrix is a matrix that has the same number of rows as columns. • For instance, here is a 3 x 3 square matrix

  3. An Identity matrix, denoted by I, has 1’s on its main diagonal and zeros everywhere else. For instance, here is the 2 x 2 identity matrix For instance, the 3 x 3 identity matrix

  4. It is noteworthy that the matrix behaves in matrix multiplication like the number 1 in number multiplication. • Just as 1 is the identity in number multiplication, (i.e., 1 times a number is the same number, for any number). • Now we see why it makes sense to call the above matrix: the 2x2 identity matrix.

  5. Inverse of a Matrix • Is there a number n such that 3 x n = 1? • You probably say ⅓ . • Is there a number n such that -1/8 x n = 1? • You probably say -8. • We say that 1/3 and -8 are the multiplicative inverses of 3 and -1/8, respectively.

  6. Definition • Similarly, when we multiply a square matrix by another matrix of the same order and obtained the identity matrix, we say that the matrices are inverses of each other. • The matrix B is the inverse of the square matrix A if AB = BA =I • The inverse of the matrix A is denoted A-1.

  7. Example #1 Show that is the inverse of the matrix

  8. Graphing calculators will be used in most cases to find the inverse of a matrix, whenever the inverse exists.

  9. Example #2 Find, if possible, the inverse of each of the following matrices Lesson: Not every nonzero matrix has an inverse.

  10. Example #3:A Telling Example Let Multiply the two matrices on the left and use matrix equality to show that the given matrix equation is equivalent to a system of linear equations.

  11. How are the coefficients of the x and y in the system of equations that you just obtained related to the square matrix in the matrix equation that was given? They are exactly the same. We thus refer to the square matrix as the coefficient matrix.

  12. Example#4:Another Telling Example Use your observations from the previous example to find matrices A, X, and B such that the matrix equation is equivalent to the system of linear equations

  13. A is the coefficient matrix

  14. Equivalence of to Systems of Linear Equations IfA, X, andBare matrices oforders n x n, n x 1, andn x 1, respectively, then to a system of n linear equations with n variables

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