Matrices and their Types

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take a grid of nine boxes, three rows and three columns, and fill every box with zero or one. How many different grids can you make? And how many of them are scalar matrices? Scalar? No idea what that means yet? Perfect! That is exactly today's lesson. And look, matrices come up often in JEE, and everything later in this unit is built on today's words. So, let's begin.

0:32A matrix is a table of numbers

So, what is a matrix? Simply a table of numbers in rows and columns, closed inside square brackets. This one has two rows and three columns. Each entry is a small a with two little numbers written below it; we will decode those in a moment. The table itself has no value; what matters is how we add and multiply tables. That is this chapter, okay? Now, in general, a matrix with m rows and n columns has order m by n. Here we read that cross sign as the word by: rows first, then columns, right? And the number of entries is m times n. So the matrix above has order two by three, and six entries. Now the short form: capital A equals small a sub i j, in square brackets, with m by n at the bottom right. The little letters below the small a are subscripts. Small a sub i j is the entry in row i, column j. Clear? Quick check. In the matrix at the top, where does small a sub two one sit? Second row, first column. Yes, exactly! Small a sub one two is first row, second column, a different place. Oh no, careful: many students read the column first and swap these two. Row first, always. Got it?

1:52Row, column, square and zero matrices

Alright, now let's give names to some shapes. Only one row? That is a row matrix, of order one by n. Only one column? That is a column matrix, of order m by one. Simple, right? When rows and columns are equal in number, order n by n, we get a square matrix. And if every entry is zero, it is the zero matrix, written with a capital O, okay? Now, picture any square matrix, and look at the entries running from the top left corner down to the bottom right. What do their two subscripts have in common? Row number equals column number! Very good. Small a sub one one, small a sub two two, and so on, up to small a sub n n; the three dots mean and so on. This line is the main diagonal, and we use it for square matrices. With me? Now add up the main diagonal, and you get the trace. The short form is t r A, read as trace of A, okay? So, trace of A equals small a sub one one, plus small a sub two two, and so on, up to small a sub n n. Again, only for a square matrix.

3:03Diagonal, scalar, identity, triangular

Now look at the five pictures below. Shaded cells may be non-zero; grey cells must be zero. The last two, symmetric, where A transpose, the letter A with a small T at its top right, equals A, and skew-symmetric, where A transpose equals minus A, with a zero diagonal, come later in this chapter. Today, we need the first three. First picture: a diagonal matrix, a square matrix with every entry off the main diagonal zero. In symbols, small a sub i j equals zero whenever i is not equal to j; the equals sign with a slash means not equal, right? Now take a diagonal matrix and make all its diagonal entries equal, the same number all along the diagonal. That is a scalar matrix. Clear? And if that same number is one? We get the identity matrix, capital I with a small n below it, of order n. So identity is a special scalar matrix, and scalar is a special diagonal matrix. Got it? Next, the second and third pictures. Upper triangular means zeros below the main diagonal: small a sub i j equals zero when i is greater than j. Lower triangular is the opposite: zeros above the diagonal, when i is less than j. Below the diagonal, the row number is bigger. With me? Now a tricky one. Is the identity matrix upper triangular? Yes, it is! Zeros below the diagonal, so upper triangular. Zeros above too, so lower triangular as well. In fact, every diagonal matrix is both upper and lower triangular. One matrix can carry many names, so in the exam, don't stop at the first one, okay?

4:46Equality of matrices

Last idea: when are two matrices equal? Capital A equals capital B means two things: the same order, and every entry matching, small a sub i j equals small b sub i j, for every i and j. So a two by three matrix can never equal a three by two matrix. Right? So tell me. If two m by n matrices are equal, how many ordinary equations do we get? m times n. Correct! One equation for every entry, and that is how we find unknown entries. Okay?

5:21Example 1: build aij = i squared minus j and find its trace

Now, Example one. Construct the three by three matrix with small a sub i j equal to i squared minus j, and find its trace. The small two up on the i means squared. Pause the video and try it first. Done? Okay, let's check. Fix a row i; the column j runs over one, two, three. So row i is i squared minus one, i squared minus two, i squared minus three. Careful: i is the row, j the column. Swap them and you build a different matrix, right? Now fill it in. Row one: i squared is one, so zero, minus one, minus two. Row two: i squared is four, so three, two, one. Row three: i squared is nine, so eight, seven, six. That's it! Now the trace. Which three numbers do we add? The main diagonal, exactly! Zero, two and six. So trace of A equals zero plus two plus six, which is eight. Clear?

6:25Example 2: counting 0-1 matrices of each type

Now, remember our puzzle from the start? Here it is as a proper question. How many three by three matrices have entries from zero and one, the set in curly brackets? How many are diagonal, how many upper triangular, and how many scalar? Pause and try. Hint: count the free entries. Done? Let's check. All nine entries are free, each with two choices, zero or one. Multiply nine twos together: two to the power nine, five hundred and twelve. That's the first half of our puzzle! Diagonal: everything off the diagonal is zero, so only the three diagonal entries are free. Two to the power three: eight. Simple! Now upper triangular. How many entries are free now? Six! Three on the diagonal and three above it. The three below must be zero. So two to the power six: sixty four. Very good! And scalar? Well, a scalar matrix is diagonal, so the answer is eight again, right? No, no, wait! Scalar means all the diagonal entries are equal. All zero gives the zero matrix, capital O, and all one gives the identity matrix, capital I, okay? Just two! Puzzle solved: five hundred and twelve grids, and only two of them scalar.

7:49Quick revision: order, which type is which, equal matrices

So, quick revision. Order m by n means m rows and n columns, and small a sub i j sits in row i, column j. Identity is scalar, scalar is diagonal, and diagonal is both upper and lower triangular. And equal matrices have the same order, and every entry equal. Now go and try the practice problems on this page. You will find them easy. All the best!