Question

Enter a 3 x 3 symmetric matrix A that has entries a11 = 2, a22 = 3,a33 = 1, a21 = 4, a31 = 5, and a32 =0

A =[ ]
and I is the 3 x 3 identity matrix, then
AI = [ ]
and
IA = [ ]

Answer 1

The given symmetric matrix A can be written as:

A =

| 2 4 5 |

| 4 3 0 |

| 5 0 1 |

The identity matrix I is:

I =

| 1 0 0 |

| 0 1 0 |

| 0 0 1 |

To find the product AI, we multiply matrix A by matrix I:

AI = A × I =

| 2 4 5 | | 1 0 0 | = | 2(1) + 4(0) + 5(0) 2(0) + 4(1) + 5(0) 2(0) + 4(0) + 5(1) |

| 4 3 0 | × | 0 1 0 | = | 4(1) + 3(0) + 0(0) 4(0) + 3(1) + 0(0) 4(0) + 3(0) + 0(1) |

| 5 0 1 | | 0 0 1 | = | 5(1) + 0(0) + 1(0) 5(0) + 0(1) + 1(0) 5(0) + 0(0) + 1(1) |

Simplifying the above multiplication, we get:

AI =

| 2 4 5 |

| 4 3 0 |

| 5 0 1 |

Similarly, to find the product IA, we multiply matrix I by matrix A:

IA = I × A =

| 1 0 0 | | 2 4 5 | = | 1(2) + 0(4) + 0(5) 1(4) + 0(3) + 0(0) 1(5) + 0(0) + 0(1) |

| 0 1 0 | × | 4 3 0 | = | 0(2) + 1(4) + 0(5) 0(4) + 1(3) + 0(0) 0(5) + 1(0) + 0(1) |

| 0 0 1 | | 5 0 1 | = | 0(2) + 0(4) + 1(5) 0(4) + 0(3) + 1(0) 0(5) + 0(0) + 1(1) |

Simplifying the above multiplication, we get:

IA =

| 2 4 5 |

| 4 3 0 |

| 5 0 1 |

Therefore, AI = IA =

| 2 4 5 |

| 4 3 0 |

| 5 0 1 |