How is the following matrix multiplication working here I do

How is the following matrix multiplication working here? I don\'t understand how the answer was derived

23-3 13 23 2-3-2-3 23 3 2 29 49 55 -2 23-3 2-3 2- 32313 -2- 9 59 49 89-29 29 5

Solution

Here we multiply and then add corresponding elements of each row of first matrix to each corresponding element of each column to get all three numbers of each row of last resulting matrix.

So this multiplication is done as :

UU^T : | (2/3)(2/3)+(-2/3)(-2/3) (2/3)(1/3)+(-2/3)(2/3) (2/3)(2/3)+(-2/3)(1/3) |

| (1/3)(2/3)+(2/3)(-2/3) (1/3)(1/3)+(2/3)(2/3) (1/3)(2/3)+(2/3)(1/3) |

| (2/3)(2/3)+(1/3)(-2/3) (2/3)(1/3)+(1/3)(2/3) (2/3)(2/3)+(1/3)(1/3) |

= | 4/9 + 4/9 2/9 - 4/9 4/9 - 2/9 |   

| 2/9 - 4/9 1/9 + 4/9 2/9 +2/9 |

| 4/9 - 2/9 2/9 +2/9 4/9 + 1/9 |

= | (4+4)/9 (2 - 4)/9 (4-2)/9 |

| (2 - 4)/9 (1 + 4)/9 (2+2) /9 |

| (4 - 2)/9 ( 2 + 2 )/9 (4 + 1)/9 |

= | 8/9 -2/9 2/9 |

| -2/9 5/9 4/9 |

| 2/9 4/9 5/9 |

This is how we get our last result.