Addition and Scalar Multiplication

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. For years in school, you have solved pairs of equations like two x plus three y equals something, and three x plus two y equals something else. Now imagine the unknowns are not numbers. Each unknown is a whole matrix, a full table of numbers. Can we still solve the pair? Sounds scary, right? By the end of this video, you will solve exactly such a pair in a few lines. And we need only two tools: adding matrices, and multiplying a matrix by a number. They look easy, but they come up often in JEE, hidden inside bigger questions. So, let's begin.

0:43Adding matrices: entry by entry

So, how do we add two matrices? Really simple: entry by entry. On screen, the small i j written below is a subscript. It means the entry in row i, column j, okay? So the i j entry of A plus B is just a i j plus b i j. But see the condition written first: same order only. Same number of rows, same number of columns. Right? Look at a two by two case. The big square brackets simply hold the entries of a matrix. Now add: a plus p, b plus q, c plus r, d plus s. Each entry meets only its partner in the same position. Nothing mixes across. Clear? Now tell me. One matrix is two by two, the other is two by three. The cross between the numbers is read as by: rows by columns, okay? Can we add these two? No, we cannot! Oh no, careful here. Some students add the four matching entries and simply copy the extra column. No, no! The third column has no partners at all, so the sum is not defined. Different orders, no sum. Got it?

1:53Multiplying a matrix by a scalar

Next, multiplying a matrix by a number. In matrix language, a plain number is called a scalar, and we usually call it k, okay? So think. In k A, which entries get multiplied by k? Only the first row? Only the diagonal? Every single entry! Yes, exactly. The i j entry of k A is k times a i j, for every row and every column. With me? So k times the matrix a, b, c, d is the matrix k a, k b, k c, k d. Nothing is left out. Simple! Now put k equal to minus one. We get minus A, where every entry changes its sign. And subtraction? Well, A minus B is just A plus minus B, okay? That makes subtraction entry by entry too, with the same order needed. Clear?

2:45The rules: algebra just like numbers

Alright, now the rules. Quick question first. For matrices of the same order, is A plus B always equal to B plus A? Yes, always! Why? Because each entry of the sum is an ordinary sum of two numbers, and numbers can be added in any order. Same reason for the brackets: A plus B, then plus C, equals A plus the bracket B plus C, always. Right? Next, the capital O here is the zero matrix: every entry zero, same order as A, okay? Adding O changes nothing. And A plus minus A gives O, because every entry cancels its own negative. Now scalars. Here k and l are both scalars, just numbers. k times the bracket A plus B equals k A plus k B, and k plus l, times A, equals k A plus l A, right? Brackets open exactly as in ordinary algebra. And the last one: k times l A equals k l, times A, got it? So you can multiply the two scalars first. In short, with addition and scalar multiples, you can do algebra on matrices just as on numbers. Keep that in mind!

4:00Topper's trick: solve them like number equations

Now the trick that toppers use. Matrix equations like these are solved exactly like equations in numbers. Two X plus three Y equals P, and three X plus two Y equals Q, where capital X and capital Y are unknown matrices, and P and Q are known matrices, all of the same order, okay? Now, if X and Y were ordinary numbers, how would you remove Y? Yes, exactly! Make the Y parts equal, then subtract. Three Q has six Y, and two P also has six Y, see? So three Q minus two P equals nine X plus six Y, minus the bracket four X plus six Y, and the Y parts cancel. We are left with only five X, beautiful! Same way for Y: take three P minus two Q, and now the X parts cancel. We get five Y, simple! Finally, multiply by one fifth. That small fraction is one over five. So X is one fifth of three Q minus two P, and Y is one fifth of three P minus two Q, clear? Every move used only addition and scalar multiples, so every move is allowed. That's it!

5:14Example 1: find 2A - 3B for two 2x2 matrices

Let's start with an easy one. Matrix A has first row one, two, and second row three, four. Matrix B has first row minus one, zero, and second row two, five. Find two A minus three B, okay? Pause the video, and try it first. Done? Okay, let's check. First, the scalar multiples. Two A: double every entry. Two, four, six, eight. Three B: triple every entry. Minus three, zero, six, fifteen. With me? Now subtract, entry by entry. First entry: two, minus three... so minus one, right? No, no, careful! The first entry of three B is minus three, not three. So it is two minus minus three, which is two plus three. The same way: four minus zero, six minus six, and eight minus fifteen. Clear? So two A minus three B is the matrix five, four, zero, minus seven. That's it! Just watch the signs, and this type gives you quick marks.

6:26Example 2: solve 2X + 3Y = P and 3X + 2Y = Q

Now, remember our puzzle from the start? Here it is. Solve: two X plus three Y equals the matrix two, three, four, zero. And three X plus two Y equals the matrix two, minus two, minus one, five. Read each matrix row by row, okay? Pause the video and try it. Use the trick we just learnt. Done? Okay, let's check. Call the first right side P, and the second one Q, just like in our trick. So five X equals three Q minus two P, right? Three Q is six, minus six, minus three, fifteen. Two P is four, six, eight, zero. Subtract, entry by entry: two, minus twelve, minus eleven, fifteen. With me? Now five Y equals three P minus two Q, the same way. Three P is six, nine, twelve, zero. Two Q is four, minus four, minus two, ten. Subtract: two, thirteen, fourteen, minus ten. Okay? So X is the matrix two, minus twelve, minus eleven, fifteen, right? Oh no, careful! That matrix is five times X, not X itself. Multiply by one fifth. So X is one fifth of two, minus twelve, minus eleven, fifteen, and Y is one fifth of two, thirteen, fourteen, minus ten. Puzzle solved, just like simultaneous equations! And a quick check, always a good habit. The two comma two entry means row two, column two. There, X has fifteen over five, that is three, and Y has minus ten over five, that is minus two. So two times three, plus three times minus two: six minus six, zero. And the first given matrix has zero there. Correct!

8:27Quick revision: same order, kA, eliminate like numbers

So, quick revision. Adding or subtracting: same order only, entry by entry. k A: multiply every single entry by k. And matrix equations: eliminate just as you do with numbers. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!