hi everyone in this video we will see how to carry out operations on matrices now if a equals this we want to evaluate a squared minus 3a plus i where i is a 2x2 identity matrix now to do this we first find a squared now a squared equals 2 3 1 5 times 2 3 1 5 so we multiply a by itself now to do this to carry out this matrix multiplication we multiply the first row by the first column here and that gives us two times two two times two plus three times one three times one and then here we have the first row times the second column so we have two times three plus three times five three times five and then for this part we multiply this row with this column to give one times two 1 times 2 plus 5 times 1 and lastly this row with this column 1 times 3 plus five times five this is equal to two times two is four plus three times one is three two times three is six plus three times five is fifteen and here one times two is two plus five times one is five one times three is three plus five times five is twenty five and this equals four plus three is seven this is 21 here two plus five is seven and three plus 25 is 28 so a squared gives this now let's look for 3a let's evaluate 3a so 3a equals 3 times a is a 2 3 1 5 now to carry out this scalar multiplication we multiply 3 by each term in the matrices each element in this matrix so three times two gives six three times three nine three times one three and three times five fifteen now the two by two identity matrix i equals 1 0 0 1 therefore a squared minus 3 a plus i equals a squared is 7 21 7 28 minus 3 a 3 is 6 9 3 15 plus i the identity matrix which is one zero zero one so this is equal to let's carry out let's um subtract this for seven minus 6 is 1 21 minus 9 is 12 7 minus 3 is 4 and 28 minus 15 is 13 plus the identity matrix one zero zero one and this is equal to one plus one we add them ten by ten one plus one is two twelve plus zero is twelve four plus 0 is 4 and 13 plus 1 is 14. and so this gives the value to this expression a squared minus 3a plus i now let's see another example if a equals this evaluates a squared minus five i minus two a where a is a two by two identity matrix so let's evaluate a squared which is equal to a times a 1 minus 3 4 and two so to carry out this multiplication we multiply the rows the first row by the first column here and so we have one times one plus minus three times four and then for this entry we have one times minus three plus minus three times two and for this entry this entry we have this row times the first column so we have four times one plus two times four two times four and then for the last entry here we have four times minus three plus two times two and this gives one times one is one minus three times four is minus twelve four times one is four two times four is eight here we have one times minus three that's a minus three minus three times two gives minus six four times minus three is minus twelve two times two is four and this is equal to one minus 12 is minus 11 minus 3 minus 6 is minus 9 4 plus 8 is 12 and minus 12 plus 4 gives minus 8. we now evaluate 5 i 5 i equals 5 times the 2 by 2 identity matrix is 1 0 0 1 and this gives so we carry out this scalar multiplication five times each entry in dimitri so five times one is five five times zero is zero five times zero is zero and five times one is five and then lastly we evaluate 2a 2a equals 2 times a a is a 1 minus 3 4 and 2 and this gives 2 times 1 to two times minus three is minus six two times four is eight and two times two is four so a squared minus five i minus 2 a equals a squared is a minus 11 minus 9 12 minus 8 then minus 5 i this is 5i 5 0 0 5 minus 2a 2 minus 6 8 4. so to do this let's uh carry out this um subtraction first so we subtract them entry by entry corresponding entry subtract corresponding entry in those two match matrices so we have minus eleven minus five that's uh minus sixteen minus nine minus zero is minus zero twelve minus zero is twelve minus 8 minus 5 gives minus 13 then minus what we have here this is 2 minus 6 8 and 4 and this gives minus sixteen minus two gives minus eighteen minus nine minus minus six that's minus nine plus six which is so minus 3 and then 12 minus 8 gives 4 and minus 13 minus 4 is minus 17. so this gives the value to this expression i hope you enjoyed this video if you have any questions you can drop them in the comments section thank you