The 2 by 2 inverse: swap, flip signs, divide by ad − bc
Slide 1 showed what an inverse does: it undoes A. Now we find it. For a 2 by 2 matrix there is a short recipe with three moves. Swap the two numbers on the diagonal, put a minus sign on the other two, and divide everything by one number, ad − bc. That number is called the determinant. Think of checking the fuel gauge before a drive: one glance tells you whether the trip can start. Here the glance is ad − bc. If it is not 0, the recipe gives A⁻¹. If it is 0, the recipe would divide by zero, and A has no inverse at all.
Lesson 3 · slide 22 × 2 inverse- A⁻¹ = 1/(ad − bc) times d−b−ca: swap a and d, flip the signs of b and c, divide by ad − bc.
- The number ad − bc is the determinant of A, written det A. For now it means one thing: it says whether a 2 by 2 inverse exists.
- A times d−b−ca is (ad − bc)I, so dividing by ad − bc leaves I.
- If ad − bc = 0 there is no inverse: the columns lie on one line.
- Always check the answer by multiplying: AA⁻¹ should be I.
The recipe
Swap a and d. Flip the signs of b and c. Divide everything by ad − bc. Here are the three moves on A = 1234, where ad − bc = 1·4 − 2·3 = −2:
Why it works
Multiply A by the swapped-and-flipped matrix, before dividing:
The off-diagonal entries cancel. Both diagonal entries are ad − bc. So the product is ad − bc times I. Divide by ad − bc and you get I. With numbers, for A = 1234:
The determinant: one number that says yes or no
The number ad − bc is called the determinant of A, written det A. In this lesson it has one job: it tells you whether a 2 by 2 matrix has an inverse. If ad − bc ≠ 0, the recipe works. If ad − bc = 0, the recipe would divide by zero, and there is no inverse.
What does ad − bc = 0 look like? Then ad = bc, and the columns ac and bd lie on one line: one is a multiple of the other. For 2412, column 2 = 2·column 1. That is the case from slide 1, where one column is a multiple of the other: two different inputs give the same output, so nothing can undo A.
The rows tell the same story. Pick any right side, say 4 and 1. As equations, 2x + 4y = 4 and x + 2y = 1 have lines with the same slope. They never cross, so this system has no solution at all:
The determinant has much more to say. Lesson 6 shows what it measures in pictures. For now, remember it as the yes-or-no number for a 2 by 2 inverse.
A worked example
Now use it. To solve 2x + y = 1 and 5x + 3y = 2, multiply b = 12 by A⁻¹:
Common mistakes
- Swapping b and c. Only the diagonal entries, a and d, trade places.
- Flipping the signs of a and d. Only b and c change sign.
- Forgetting to divide by det A. The worked example hides this, since det A = 1. The recipe demo at the top, A = 1234 with det A = −2, does not hide it: there you must divide by −2.
A bigger example: a 3 × 3 built from a 2 × 2
The recipe is only for 2 by 2. But it still helps when the top-left 2 by 2 corner of a 3 by 3 matrix has only zeros beside it and below it, so it never mixes with the third row and column:
The top-left corner is the A from the worked example, so its inverse goes in the same corner, and the 4 is undone by ¼. For a general 3 by 3, or a 1000 by 1000, there is no short recipe worth using: the inverse comes from elimination, on slide 4.
Next: We can now undo one matrix. What if A and B are done one after the other? Slide 3 finds the inverse of a product AB, and the order turns out to flip.
Examples
Example 1 · Determinant 1
A = 2312, det A = 4 − 3 = 1.
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Example 2 · Determinant −2
A = 1234, det A = 4 − 6 = −2.
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Example 3 · No inverse
A = 2412, det A = 4 − 4 = 0.
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Example 4
Invert 4231.
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det = 4 − 6 = −2. Inverse = −½·1−2−34 = −0.511.5−2.
Example 5
For which t is t14t invertible?
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det = t² − 4. Invertible for every t except t = 2 and t = −2.
Questions and comments
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