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Unit 14: Matrices

Grids of numbers that add, multiply, and invert by their own peculiar rules, and how they solve a system in one shot.

8 topics · 30 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Matrix Dimensions, Addition, and Subtraction

  1. How it worksHow do you add two matrices by combining the entries that sit in matching positions?
  2. Worked exampleFind the sum of the matrices [[2, -1], [3, 5]] and [[4, 0], [-2, 6]].
  3. Common mistakeWhy can't you add a 2 by 3 matrix to a 3 by 2 matrix, even though they hold the same number of entries?
Your turn1 of 2
Given the matrices [[7, 6], [8, 3]] and [[2, 3], [7, 1]], find the entry in row 1, column 2 of their sum.

Scalar Multiplication of Matrices

  1. How it worksHow does a scalar distribute across every single entry when it multiplies a matrix?
  2. Worked exampleFind 3 times the matrix [[1, -2], [0, 4]].
  3. Worked exampleFind negative 2 times the matrix [[5, -1], [0, 3]].
Your turn1 of 2
Find the entry in row 2, column 1 of 7 times the matrix [[6, 8], [3, 2]].

Multiplying Two Matrices

  1. How it worksHow does each entry of a matrix product come from a row of one matrix meeting a column of the other?
  2. Worked exampleFind the product of the matrices [[1, 2], [3, 4]] and [[5, 6], [7, 8]].
  3. Common mistakeWhy doesn't matrix multiplication give the same result if you swap the order of the two matrices?
  4. Going furtherWhat has to be true about two matrices' dimensions before they can even be multiplied together?
  5. Ties togetherHow does matrix multiplication needing compatible dimensions mirror composing two functions needing compatible domains?
Your turn1 of 2
Find the entry in row 1, column 1 of the product of [[7, 6], [8, 3]] and [[2, 3], [7, 1]].

The Identity Matrix

  1. How it worksHow does multiplying any matrix by the identity matrix leave it completely unchanged?
  2. Worked exampleFind the product of the matrix [[3, 5], [2, 1]] and the 2 by 2 identity matrix, and confirm the result.
  3. Common mistakeWhy isn't a matrix full of ones the same thing as the identity matrix?
Your turn1 of 2
Find the entry in row 2, column 2 of the product of the matrix [[7, 6], [8, 3]] and the 2 by 2 identity matrix.

The Determinant of a 2x2 Matrix

  1. How it worksHow does ad minus bc become the determinant of a 2 by 2 matrix?
  2. Worked exampleFind the determinant of the matrix [[4, 3], [2, 5]].
  3. Common mistakeWhy does a determinant of zero mean a matrix has no inverse at all?
  4. Going furtherWhat happens to a matrix's determinant when you swap its two rows with each other?
Your turn1 of 2
Find the determinant of the matrix [[7, 6], [8, 3]].

The Inverse of a 2x2 Matrix

  1. How it worksHow does swapping, negating, and dividing by the determinant produce the inverse of a 2 by 2 matrix?
  2. Worked exampleFind the inverse of the matrix [[2, 1], [5, 3]].
  3. Common mistakeWhy doesn't the inverse of a matrix mean flipping every entry into its reciprocal?
  4. Going furtherWhat happens to the inverse formula's result when the determinant itself works out to a fraction?
  5. Ties togetherHow does needing a nonzero determinant for a matrix inverse connect to needing a nonzero denominator for a reciprocal?
Your turn1 of 2
Find the entry in row 1, column 1 of the inverse of the matrix [[8, 2], [3, 6]].

Solving a System of Equations Using a Matrix Inverse

  1. How it worksHow does multiplying both sides of a matrix equation by the coefficient matrix's inverse isolate the variables?
  2. Worked exampleSolve the system 2x + y = 8 and 5x + 3y = 19 by writing it as a matrix equation and using the inverse.
  3. Common mistakeWhy doesn't a system with a coefficient matrix of determinant zero have exactly one solution?
  4. Ties togetherHow does solving a system with a matrix inverse compare to the elimination method used earlier on nonlinear systems?
Your turn1 of 2
Solve the system 8x + 2y = 2 and 3x + 6y = 5 by writing it as a matrix equation and using the inverse. Find x.

Matrices as Geometric Transformations

  1. How it worksHow does a 2 by 2 matrix act on every point of a shape to rotate it around the origin?
  2. Worked exampleFind where the point (3, 1) lands after being transformed by the matrix [[0, -1], [1, 0]].
  3. Ties togetherHow does multiplying a complex number by i connect to multiplying a point by this same 90 degree rotation matrix?
Your turn1 of 2
Find the x-coordinate of the point (7, 6) after being transformed by the matrix [[0, -1], [1, 0]].