The Inverse of a Matrix

Transcript

0:00Introduction

Hello everyone! Okay, quick question first. Three times one third is one, right? Every non-zero number has a partner, its reciprocal, and together they give one. For matrices, the identity matrix plays the role of one. So, does every non-zero matrix have such a partner? Most students say yes. But no! Even a matrix with no zero entry can fail. By the end of this lesson, you will know the one number that decides it. And inverses come up often in JEE questions. So, let's begin.

0:36What is an inverse?

So, what exactly is an inverse? Take a square matrix A, and suppose some matrix B gives A B equal to I, and B A equal to I too, where capital I is the identity matrix. Both orders, because matrix multiplication is not commutative. Right? Then this B is unique, and we write it as A with a small minus one at the top, read as A inverse. Careful: it is just a name; it does not mean one over A, because we never divide by a matrix. Got it? And such a matrix is called invertible, or non-singular. Now let's see which matrices are invertible.

1:16When does the inverse exist?

Recall from the last lesson: matrix A times adj A equals determinant of A, times I, the identity matrix. The letters a d j are short for adjoint, and the straight bars mean determinant, okay? But we want just the identity on the right side. So what should we do? Divide by the determinant, yes, exactly! Multiply adj A by one over determinant of A, and matrix A times this, in round brackets, is exactly the identity. But we can divide only when determinant of A is not zero; the slashed equals sign means not equal. Right? So the inverse of A is one over determinant of A, times adj A: our formula. And the double-headed arrow means if and only if: matrix A is invertible exactly when its determinant is non-zero. That is the one number from our puzzle! Got it? See this matrix from the last lesson, in square brackets, with rows one, two and two, four. No zero entry. But its determinant is four minus four, zero. So it has no inverse; it is singular. Puzzle solved!

2:26The 2x2 inverse in seconds

Now the two by two case; the cross in the heading is read as by. For the matrix with rows a, b and c, d, the determinant is a d minus b c. Its inverse is one over a d minus b c, times the matrix with rows d, minus b and minus c, a. Okay? That second matrix is just the adjoint, by the shortcut you know: swap the diagonal entries, a and d, and change the signs of b and c. Quick, right? Many students stop right there. Is the adjoint itself the inverse? No, no, careful! You must still divide by a d minus b c, which must not be zero. That slip is very common. With me?

3:11Rules for inverses

Now the rules. Let A and B be invertible, of the same order. Is the inverse of A B just A inverse times B inverse? No! The order reverses: the inverse of A B is B inverse times A inverse, the reversal law. Check it: in the middle, B times B inverse gives I, and then A times A inverse gives the identity again. The wrong order is the classic trap here. Clear? Three quick ones. The inverse of A inverse is A itself. The small capital T means transpose, and the inverse of A transpose is the transpose of A inverse. And the determinant of A inverse is one over determinant of A, because determinants multiply, and the identity has determinant one. Okay? For a non-zero number k, the inverse of k A is one over k, times A inverse. And the inverse of A to the power n is A inverse, to the power n. Got it? Last, cancellation. If A is invertible and A B equals A C, then B equals C; the arrow means implies. Just multiply both sides on the left by A inverse. But if determinant of A is zero, you cannot cancel like this. Right?

4:31Topper's trick: the inverse from an equation

Now a topper's trick. Suppose matrix A satisfies A squared, minus five A, plus seven I, equals capital O, the zero matrix. Keep the seven I alone on one side. What can you take common from the rest? Yes, matrix A! Seven I equals five A minus A squared, which is A times, in brackets, five I minus A, okay? Oh no, careful: write five I, not just five. You cannot subtract a matrix from a number. With me? Now divide by seven: matrix A times one seventh of, five I minus A, gives the identity, and so does the other order. So the inverse of A is one seventh of, five I minus A, with no cofactors! Right? One condition: the constant term, seven I here, must be non-zero. Otherwise there is no identity to make. Clear?

5:26Example 1: inverse of a 2x2 matrix with determinant 1

Alright, Example one, an easy one. Find the inverse of the matrix with rows three, five and one, two. Pause the video, and try it first. Done? Okay, let's check. Call this matrix A, and find its determinant: three times two, minus five times one; the dot means times. Six minus five: one. Not zero, so the inverse exists. Right? Now swap three and two on the diagonal, change the signs of five and one, and divide by the determinant, one. So the inverse has rows two, minus five and minus one, three. That's it! Always check: matrix A times its inverse. Row one: three times two plus five times minus one is one; three times minus five plus five times three is zero. Row two gives zero and one. The identity matrix. Perfect! Got it?

6:26Example 2: inverse of a 3x3 matrix from its cubic equation

Now Example two, a step harder. Matrix A is three by three, with rows two, minus one, one; minus one, two, minus one; and one, minus one, two. It satisfies A cubed, minus six A squared, plus nine A, minus four I, equals the zero matrix. Find its inverse. Pause and try. Hint: use the topper's trick. Done? Okay, let's check. Keep the constant term alone: minus four I moves to the right side and becomes plus four I, okay? Now take A common on the left: A times, in brackets, A squared minus six A plus nine I, equals four I, with nine I, not nine! Divide by four: the inverse of A is one fourth of that bracket. Clear? How much work is left now? Do we need to compute A cubed? No, only A squared! Row one of A times column one: four plus one plus one, six. Row one times column two: minus two, minus two, minus one, so minus five. The rest go the same way: six on the diagonal, five or minus five elsewhere. With me? Now A squared, minus six A, plus nine I: on the diagonal, six minus twelve plus nine, three. Where A has minus one: minus five plus six, one. Where A has plus one: five minus six, minus one. So the inverse is one fourth of the matrix with rows three, one, minus one; one, three, one; and minus one, one, three. Imagine nine cofactors instead! That is the power of this trick.

8:12Quick revision: when it exists, formula, 2x2, rules

So, quick revision. The inverse exists exactly when the determinant is non-zero, and then it is adj A over that determinant. For two by two: swap the diagonal, change the other two signs, and divide by a d minus b c. The inverse of a product reverses the order, and an equation with a non-zero constant term hands you the inverse directly. Now go and try the practice problems on this page. All the best!