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MA1506CHAP5 - CHAPTER 5 MATRICES AND THEIR USES 5.1 What is...

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CHAPTER 5 MATRICES AND THEIR USES 5.1 What is a Matrix? A system of linear algebraic equations in two vari- ables might look like this: 2 x + 7 y = 3 4 x + 8 y = 11 LINEAR because it just involves constant multiples of x and y , no x 2 , no sin( y ), etc. ALGEBRAIC because no differentiation. It’s cool to write these systems using the following notation: 2 7 4 8 ‚ • x y = 3 11 . Here x y and 3 11 are familiar - they are VEC- 1
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TORS. But 2 7 4 8 is something new, called a MA- TRIX. We say that the PRODUCT of 2 7 4 8 with x y gives you 3 11 . Every matrix has ROWS and COLUMNS. In this case, the rows are [2 7] and [4 8] and the columns are 2 4 , 7 8 . We call [2 7] a ROW VECTOR and 2 4 a column vector. We say that 2 7 4 8 is a 2 by 2 matrix since it has two rows and two columns. You can regard [2 7] as having one row and 2 columns, etc. You can also have 3 by 3 matri- ces like 1 7 9 7 8 2 4 10 12 or even 2 by 3 matrices like 1 2 4 6 8 9 two rows, three columns. 2
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A general 3 by 3 matrix can be written as a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 so a ij is the number in the i -th row and j -th column, Note a ij 6 = a ji usually! Engineers and physicists like to talk about “the ma- trix a ij ”. Strictly speaking, they mean “the matrix with entries a ij ” but we will talk in this sloppy way too! In the same way, any column vector can be written as u 1 u 2 u 3 . 5.2 Matrix Arithmetic [a] Addition and Subtraction. Just add up or subtract the entries, as you would for a vector. 3
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1 2 4 8 + 7 3 6 9 = 8 5 10 17 1 2 4 8 - 7 3 6 9 = - 6 - 1 - 2 - 1 In general, if a ij and b ij are matrices (both m by n , that is, both have m rows and n columns) then the sum is a ij + b ij and the difference is a ij - b ij . [b] Multiplying By a Number. Just multiply every entry, as you would for a vector. 2 · 1 2 4 8 = 2 4 8 16 . The product of the number c with the matrix a ij is c · a ij . [c] Transposition. If you take a matrix and SWITCH THE FIRST ROW INTO THE FIRST COLUMN, second row into second column, and so on, the result is called the TRANSPOSE. We write 1 2 4 8 T = 1 4 2 8 . 4
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1 7 9 6 8 2 4 10 12 T = 1 6 4 7 8 10 9 2 12 1 2 4 6 8 9 T = 1 6 2 8 4 9 a 11 a 12 a 21 a 22 T = a 11 a 21 a 12 a 22 and by looking at this example you can see a T ij = a ji the order of the indices is reversed. Notice that ( a ij ) T · T = ( a ji ) T = a ij ( a ij + b ij ) T = a ji + b ji = a T ij + b T ij ( c a ij ) T = c a ji = c ( a ij ) T . [d] Multiplying Matrices. We started by declaring that it was cool to write 2 x + 7 y = 3 4 x + 8 y = 11 as 2 7 4 8 ‚ • x y = 3 11 . Clearly this 5
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is a way of saying that the vector 2 x + 7 y 4 x + 8 y equals 3 11 , so 2 7 4 8 ‚ • x y = 2 x + 7 y 4 x + 8 y . Notice that ROWS of 2 7 4 8 multiply the COLUMN x y . We adopt this as our GENERAL RULE: ROWS MULTIPLY COLUMNS!
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