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Precalculus 9-12 / Week 11 / Thursday
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Week 11 · Matrices

Thursday

Undoing a matrix
// The lane card grid
⏱ about 20 min

Thursday: Undoing a Matrix

"The stretched hull is [2 0; 0 1] times the old one," Comet says. "Can a matrix stretch it back?"

"Halve the x," Wren says, writing [1/2 0; 0 1]. "Multiply the two and you get the identity. That is the inverse."

"What about this one?" Comet writes [2 4; 1 2]. "Determinant 0."

Nova projects the hull squashed flat onto a single line. "Would you like a hint? The area factor is zero."

"Every point lands on one line," Wren says. "You cannot tell which point it came from. No inverse."

"So the determinant decides," Comet says. "Not zero, there is an inverse. Zero, there is none."

"And the inverse solves the lane cards," Wren says. "2x + y = 7 and x + 3y = 11, as one matrix sentence."

"What do you notice? It is the winch rule run backwards, for two unknowns at once."

The inverse of a 2 by 2 matrix

The inverse of A, written A⁻¹, is the matrix with A × A⁻¹ = I and A⁻¹ × A = I. It undoes A.

For A = [a b; c d], swap a and d and flip the signs of b and c. Then divide every entry by the determinant ad - bc.

If the determinant is 0, dividing is not allowed, and there is no inverse. The matrix squashes the plane onto a line.

So: the determinant is not zero exactly when the inverse exists. One number answers the question.

A solved problem: undoing the stretch

  1. A = [2 0; 0 1]. Determinant: 2 × 1 - 0 × 0 = 2, not zero, so an inverse exists.
  2. Swap a and d: [1 0; 0 2]. Flip the signs of b and c: still [1 0; 0 2], because they are zero.
  3. Divide every entry by 2: A⁻¹ = [1/2 0; 0 1].
  4. Check: [2 0; 0 1] × [1/2 0; 0 1] = [1 0; 0 1], the identity.
  5. Now [2 4; 1 2]: determinant 2 × 2 - 4 × 1 = 0. No inverse. Every point lands on the line y = x/2.

A system as one matrix sentence

The lane cards: x short heats and y long heats. A short heat uses 2 blue cards and 1 white card; a long heat uses 1 blue and 3 white.

The board holds 7 blue cards and 11 white cards. So 2x + y = 7 and x + 3y = 11.

Write it as A[x; y] = b with A = [2 1; 1 3] and b = [7; 11]. Row 1 of A times [x; y] is the first equation.

Multiply both sides by A⁻¹: [x; y] = A⁻¹b. The determinant of A is 5, so A⁻¹ exists and the answer is unique.

A⁻¹ = [3/5 -1/5; -1/5 2/5], and A⁻¹b = [2; 3]. So x = 2 short heats and y = 3 long heats.

Check: 2 × 2 + 3 = 7 and 2 + 3 × 3 = 11. Both cards balance.

Why AB can differ from BA

Yesterday a stretch and a flip moved the hull. Here is why doing them in a different order gives a different matrix. Read it, then order it.

  1. Let S = [2 0; 0 1], the stretch across, and F = [0 1; 1 0], which swaps x and y.
  2. Take the point (1, 0). Apply F first: it becomes (0, 1). Then S: it stays (0, 1).
  3. So SF, which means F first and then S, sends (1, 0) to (0, 1).
  4. Now apply S first: (1, 0) becomes (2, 0). Then F: it becomes (0, 2).
  5. So FS sends (1, 0) to (0, 2). The two images differ, so SF and FS cannot be the same matrix.
  6. Multiplying confirms it: SF = [0 2; 1 0] and FS = [0 1; 2 0]. Matrix multiplication is not commutative.

It is still associative, (AB)C = A(BC), and distributive, A(B + C) = AB + AC. Only the order of a product is special.

THE PROOF, IN ORDER
  • ?S sends (1, 0) to (2, 0), and then F sends it to (0, 2)
  • ?Therefore SF = [0 2; 1 0] and FS = [0 1; 2 0] differ
  • ?So SF sends (1, 0) to (0, 1)
  • ?So FS sends (1, 0) to (0, 2), a different point
  • ?F sends (1, 0) to (0, 1), and then S leaves (0, 1) alone
  • ?Let S = [2 0; 0 1] (stretch) and F = [0 1; 1 0] (swap x and y)
WHY THIS EXERCISEOne point landing in two places is enough to prove two matrices are different.
DETERMINANTS AND INVERSES
  • Read the question.
  • Tap your answer.
What is the determinant of [3 1; 2 4]?
What is the determinant of [2 4; 1 2]?
What is the inverse of [2 1; 1 1]?
What is the inverse of the lane card matrix [2 1; 1 3]?
SOLVE WITH A MATRIX
  • Read the question.
  • Tap your answer.
The lane cards give 2x + y = 7 and x + 3y = 11. Write A[x; y] = [7; 11] and solve with A⁻¹.
Two heats share 10 boats and one has 2 more: x + y = 10 and x - y = 2. Write A[x; y] = [10; 2] and solve with A⁻¹.
A matrix has determinant 0. Which statement about it is true?
The number ad - bc for the matrix [a b; c d] is called the this. Type one word.
What is the determinant of the lane card matrix [2 1; 1 3]? Type the number.
A matrix times its inverse gives the this matrix. Type one word.
StatementTrue or false?
A 2 by 2 matrix has an inverse exactly when its determinant is not zero.?
2 × 2 - 4 × 1 = 0?
2x + y = 7 and x + 3y = 11 can be written as [2 1; 1 3][x; y] = [7; 11].?
To solve A[x; y] = b, multiply both sides by A.?
SF and FS are always the same matrix.?
WHY THIS EXERCISEThe matrix sentence is the system. The inverse is the solving step. The determinant says whether it works.
Try it
Write a 2 by 2 matrix of your own with a nonzero determinant. Find its inverse and multiply to check for I.
Then change one entry so the determinant is 0. Say what happens to the plane.

Sharp thinking. Tomorrow matrices leave the shed: tables everywhere, a review, and your log.

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