Question

2. Partitioned matrices A matrix A is a (2 x 2) block matrix if it is represented in the form [ A 1 A2 1 A = | A3 A4 where each of the A; are matrices. Note that the matrix A need not be a square matrix; for instance, A might be (7 x 12) with Aj being (3 x 5), A2 being (3 x 7), A3 being (4 x 5), and A4 being (4 x 7). We can imagine creating a (2 x 2) block matrix by dividing the array into four pieces using a horizontal line and a vertical line. Now suppose B is also a (2 x 2) block matrix given by B = [ By B2 1 B3 B4 Finally, let us suppose that the product AB can be formed and that B has been partitioned in a way such that the following matrix is defined: AjB1 + A2B3 A1B2 + A2B4] A3B1 + A4B3 A3B2 + A4 B4 It turns out that the product AB is given by this block matrix. That is, if all the submatrix products are defined, then we can treat the blocks in a partitioned matrix as though they were scalars when forming products. It is tedious to prove this result in general, so we ask you to illustrate its validity with some randomly chosen matrices. a) Using the MATLAB command round (10*rand (6, 6)) generate two randomly selected (6 x 6) matrices A and B. Compute the product AB. Then write each of A and B as a block matrix of the form A = [ A1 A2 | A3 A4 A4 | B= BB2 | B3 B4

Answer 1

a) the random generated matrices are:-

a) The randomly generated matrices are A-61 2 61 281, A, AAA 7 7 3 8 6 8 ** o suo 36 LA AY AT Als of 63 8 2 0 1 2 Aigt Au PLAAN 0 2 2 1 6 ** Ang Aza Ay Ayal 1 8 5 & A5, A3, A4 - it omo- 8- o a . 7 4. Rege | o o º 3 aq 4 5 1 2 0 0 its o 5 To a 2 1 6 8 89 Bio & 3 5 11ez Bin as a BA BIN 9 8 1 BB20 B2 28 3 6 5 Les 83 86 227 1831 82 83 Po Bra

AB+ A ts BetaBata Bis +36) BAC +4_3179 But2482 +82 +) L Lear+A+Bato B AOBA +Anbu tf283] ... abbit f 14 B7+A SESTAI BAT ALABr + Air B . . Art A20 B7 + A2, B13 + P2289 A23BET Azy B] - AB, HALBt & A27B13 + A Big + A29 mot As obs] .. A 3R + A32 Byt A33B13+ Alu Big Ass But A36 B31]. in MOIS = 72 114 & 65 126 104 118 No. of Addi. Sheets

1 2 6 1 281 O 090 71 7 7 3 8 6 8 3 9 9 7 9 4 0 5 4 0 3 6 1 4 5 1 6 8 3 8 2 0 1 2 1 2 0 0 3 1 5 0 4 2 2 16 5 0 5 981 18 5 0 4 8 6 9 2 3 6 5 1x0+2 x 3 +6x1+1 2+2 x 5 +8x61x0+2 x 9 + 6 x 4+1x0+2x+8 x91x9+2 x 9 + 6 x 5+1x0+2 x 5 + 8x2 1x0+2 x 7+6x1+1 X 3+2 x 9 +8x31x7+2 x 9 +6 x 6+1x1+2 x 8 + 8 x6 1x1+2x4+6x8+2x5+2x1+8 x 5 7% 0 +7x3 + 3x1 +82 + 6 x 58 x 6 7x0+ 7 x 9 + 3 X 4+8X0+ 6 x 0+8x97x97x93x58xO+6x58x27x0+7X 7+3x18x3 +6x9 +8x37x7+7 x 9 + 3x6 +8xl+6x8+8x67x17x4+3 8+8x5 + 6xl+8 x 5 OxO+5x3+4x1+0 x 2+3 x 5 +6x6 OxO+5 x 9 +4x4+OX 0+3x0+6x9 Ox 9 +5 x 9 + 4x5 + 0x0+3 x 5+ 6x2 OxO+5 x 7+ 4x10x3+3 x 9 + 6 x 3 Ox 7+5 x 9 +4x6+Oxl+3 8+6x6 Oxl+5x 4+4 x 8+0x5+ 3xl+ 6 x 5 3x0+8x3+2x10x2+1 5+2 x 6 3 x 0+8 x 9+2x4+0x0+10+2x93x9+8X9+2 5+ 0x0+1x5+2x2 3x0+8x7+2x10x3+1 x 9+2 x3 3x7+8 x 9+2 x 6+0x1+1x8+2x63x1+8 x 4+2x8+0x5+1x1+2 x 5 Ox0+4x3+2x1+2 x 2 +1x5+6x6 Ox0+4x9+2x4+2 x 0+10+ 6 x 9 x 9 +49 +2x5+2x0+1x 5+6X2 OX0+4 x 7+2 x 1+2 3+1x 9 + 6 x 30X7 +4 x 9 +2 x 6+2xl+1x8+6x6 Oxl+4x4+2x8+2x5+2xl+ 6 x 5 | 1x0+8X3 +5X1 + 0x2 + 4 X 5+ 8 x 6 x 0 +8 x 9 +5 X 4+OXO +4 0 +8x91x 9 +8 x 9 +5 x 5 +0x0+4 x 5+ 8x2 1x0+8 x 7+ 5 x 1+0 X 3+4 9+8x31x 7+8 x 9 +5X 6+0x1+4 X 8 + 8 x 6 x 1+8X4+5 8 +0 X 5+4x +8 5 72 114 83 65 126 1041 118 147 187 154 234 145 70 115 92 84 129 85 43 98 118 73 125 62 59 98 63 63 94 73 97 164 142 121 189 117]
b)

1st example:-2nd example:-

1 2 6 1 2 8 10 0 9 0 7 1 7 7 3 8 6 8 3 9 9 7 9 4 0540361 4 5 1 6 8 3 8 2 0 1 2 2 0 0 3 3 5 0 4 2 2 6 544 0 5 5 8 1 18 5 0 4 86 92 36 51 1x0+2 x 3 +6x1+1x2+2 x 544 + 8 x 6 1x0+2 x 9 + 6 x 4+1x0+2x+8x9 1x 9+2 x 9 +6x5 +1x0+2 x 5 +8x2 1x0+2 x 7+6x1+1x3+2 x 5 +8 x 31 x 7+2 x 9 +6 x 6+1 x 3 + 2 x 8+8 x6 1: 7x0+7x3+3x1+8 x 2 + 6 x 544 + 8 x 6 7x0+ 7 x 9 +3 x 4 + 8 x 0+6 0 +8x9 7 x 9 + 7 x 9 + 3 x 5 +8 x 0 + 6 x 5 +8x27x0+7 x 7+3 x 1 8 x 3 + 6 x 5 +8 x 3 7 x 7+ 7 x 9 + 3 x 6+8 x 3 + 6 x 8 8 x 6 7 0x0+5x3+4x1+0x2+3 x 544 +6x6 Ox 0+5 x 9 + 4 x 4+0x 0+3x0+6x9 Ox 9+5 x 9 +4x5+0+3 x 5+6 2 0x0+5 x 7+ 4x1+0x3+3 x 5+ 6 x 3 Ox 7+5 x 9+4 x 6+0x3+3 8+6 6 0 3x0+8X 3+2x1+OX 2+1 x 544 +2 x 6 3X0+8x9+2x4+0xO+1x0+2x93x9+8 x 9 +2 x 50 x0+1x5+2x2 3x0+8 x 7+2 x 1+OX 3+1x5+2x3 3x 7+8 x 9 +2 x 6+0x3+1x8+2x6 3 0x0+4x3+2x1+2 x 2+1 x 544 + 6 x 6 Ox0+ 4 x 9 +2 x 4+2 x 0+10+6x9 0 x 9+4 9+2 5+2 x 0+1x 5+6 x 2 Ox0+4 x 7+2x1+2x3+1 5+6 x 30 x 7+ 4 x 9+2 x 6+2x3+1x8+6x6 0 1 x 0 +8 x 3 +5X1+0 2 + 4 x 544+ 8 x 6 x 0 + 8x 9 +5X 4+0X0 +4 0 +8 x 9 x 9 +8 x 9 +5 x 5 +OXO +4 5 + 8x2 1 x 0 +8 x 7 +5x1+0X 3+4 X 5+8x31x7+8 9 +5 X 6+0x3+4x8+8X6 1150 114 83 57 128 104 3352 147 187 130 250 145 1687 115 92 72 129 85 582 98 118 69 125 62 598 98 63 59 98 73 2253 164 142 105 189 117

3rdexample:-

100 7 3 39 0 5411 45 3 8 2 2 0 0 4 2 544 05 8 5 1 6 9 2 Result [1 2 61 28 To 0907 11 7 7 3 8 6 813 99794 05 4 0 36 1 4 5 1 6 8 3 8 20 i 21 | 2 00335 0 4 2 2 1 6 544 0 5 5 8 1 18 5 0 4 86 92 36 51 1x0+2 x 3 +6x1+1x2+2 x 544 + 8 x 6 1x0+2 x 9 + 6 x 4+1x0+2x+8x9 1x 9+2 x 9 +6x5 +1x0+2 x 5 +8x2 1x0+2 x 7+6x1+1x3+2 x 5 +8 x 31 x 7+2 x 9 +6 x 6+1 3+2 x 8+8 x 6 lxl+2 x 4 + 6 x 8+1x5+2x1+8 x 5 7 x 0 +7X3 + 3x1+8 x 2 + 6 x 544 + 8x 6 7x0+7x9 +3 x 4+80+60+8x97x97x9+ 3x5 + 8x0+ 6 x 5+8x27x0+7 x 7+3 xl+8X3 + 6 x 58 x 3 7 x 7 + 7 x 9 +3 6+8x3+6x8+8 x 6 7xl+7x4+3 x 8+8x5+6x1+8 x 5 0x0+5x3+4x1+0x2+3 x 544 + 6 x 6 0x0+5 x 9 + 4 x 4 +0x 0+3x06x9 Ox 9+5 x 9+ 4x50x0+3 x 5+6 x 2 Ox0+5 x 7+ 4x1+0x3+3 x 5+6 x 30 x 75 x 9+ 4 x 6+0x3+3x8+6x6 Ox1+5 x 4+4x8+0x5+3x1 + 6 x 5 3 x 0+8X3 + 2x1+0 x 2 + 1 x 544 +2x6 3X0+8 x 9 +2x4+OX0+1x0+2x93 x 9 + 8 x 9+2 x 5 + OXO +1x5+2x2 3x0+8 x 7+2x1+OX3+1x5+2x3 3x 7 + 8 x 9 +2 x 6+OX 3+1x8+2x6 3x1+8x4 + 2 x 8 +0x5+1x1 + 2 x 5 Ox0+4x3+2x1+2 x 2+1 x 544 +6x6 Ox0+ 4 x 9 +2 x 4+2 x 0+10+6 x 90 x 9+4 9+2 5+2 x 0+1 5+6 x 2 Ox0+ 4 x 7+2 x1+2x3+1 5+ 6x3 0 x 7+ 4x 9+2 x 6+2 x 3+1x8+6 x 6 Oxl+4 x 4+2x8+2x5+1x1+ 6 x 5 11 x 0 +8 x 3 +5 1 +0x2+4 x 544 + 8 x 6 1 x 0 +8 x 9 +5 X 4+OX0+4X0 + 8x 9 1 x 9 +8 x 9 +5 x 5 +0X0 +4 X 5 +8x21x0+8 x 7+5 x 1+0x3 +4 X 5 +8 x 3 1 x 7+8 x 9 +5 X 6+OX 3+4 X 8 + 8 x 6 lxl+8X4+5 X 8 +0 X 5+4x1+8 x 5 1150 114 83 57 128 104 3352 147 187 130 250 145 1687 115 92 72 129 85 582 98 118 69 125 62 598 98 63 59 98 73 2253 164 142 105 189 117 Activate Windows Go to Settings to activate Windows.

Result 1x0+2x31x0+2 x 91 7x0+7X3 7x0+7 x 9 0x0+5 X 3 Ox0+5 x 9 3x0+8x3 3x 0 +8 x 9 OX0 + 4 X 30X0 +4x9 1 x 0 +8x31x0+8 x 9

c)

Result 0 0 0 1 111x0+2 x 3+0x0 1x0+2 x 9 +0x0 1x0+2 x 0 + 0x01.16 18 01 7701 17X0+7X 3+0x0 7x0+ 7 x 9 +0X0 7x0+7X0+0x 2163 01 LOOD

error is triggered when no of columns in first matrix is not equal to no of rows in 2nd matrix..