• Matrices
• Types
• Adding
• Scalar M.
• Multiplying
• Inverse
• Rank
• Equations

Two matrices A and B can be mulitplied together if the number of columns of A is equal to the number of rows of B.

M_{m x n} x M_{n x p} = M_{m x p}

The element, c_ij, of the product matrix is obtained by multiplying every element in row i of matrix A by each element of column j of matrix B and then adding them together.

Properties of Matrix Multiplication

Associative:

A · (B · C) = (A · B) · C

Multiplicative Identity

A · I = A

Where I is the identity matrix with the same order as matrix A.

Not Commutative:

A · B ≠ B · A

Distributive:

A · (B + C) = A · B + A · C

Examples

1. Given the matrices:

Calculate:

  A x B;     B x A

 

2.Given the matrices:

Determine if the following multiplications are possible:

1.(A^t · B) · C

(A^t_{3 x 2} · B_{2 x 2}) · C_{3 x 2} = (A^t · B)_{3 x 2} · C_{3 x 2}

  The multiplication is not possible because the number of columns, (A^t · B ) does not coincide with the numbers of rows of C.

2.(B · C^t ) · A^t

(B_{2 x 2} · C ^t_{2 x 3} ) · A^t_{3 x 2} = (B · C)_{2 x 3} · A^t_{3 x 2} =

=(B · C ^t · A ^t ) _{2 x 2}

3Determine the dimension of M so that the multiplication is possible: A · M · C

A_{3 x 2} ·  M_{m x n} ·  C_{3 x 2}            m = 2

4Determine the dimension of M if C ^t · M is a square matrix.

  C ^t_{2 x 3} · M_{m x n}                     m = 3     n = 3

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