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Unit 3: Matrix Algebra, Inverses, and LU Factorization

How matrices multiply, when they can be undone, and how elimination itself gets packaged into the factorization A = LU.

14 topics · 49 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-Vector Multiplication, Two Ways

  1. How it worksHow does each entry of Ax come from the dot product of one row of A with the vector x?
  2. How it worksHow does Ax also come out as a combination of the columns of A weighted by the entries of x?
  3. Worked exampleFind the product of the matrix [[1, 2], [3, 4]] and the vector (5, 6).
  4. Common mistakeWhy isn't matrix-vector multiplication just multiplying the matching entries together?
Your turn1 of 2
Find the first entry of the product of the matrix [[3, 2], [7, -4]] and the vector (-7, -5).

Matrix-Matrix Multiplication

  1. How it worksHow does each entry of AB come from a row of A dotted with a column of B?
  2. Worked exampleFind the product of the matrices [[1, 2], [0, 1]] and [[3, 0], [1, 2]].
  3. Common mistakeWhy doesn't multiplying two matrices entrywise, the way you'd add them, count as matrix multiplication?
  4. Going furtherWhat happens to the size of the product when a 2 by 3 matrix multiplies a 3 by 4 matrix?
Your turn1 of 2
Find the (1,1) entry of the product of [[3, 2], [7, -4]] and [[-7, -5], [3, -8]].

Matrix Multiplication Is Not Commutative

  1. Common mistakeWhy doesn't AB equal BA the way ordinary number multiplication does?
  2. Worked exampleFind AB and BA for A equal to [[1, 1], [0, 1]] and B equal to [[1, 0], [1, 1]], and compare them.
  3. How it worksHow does the row-times-column rule explain why changing the order changes which entries get combined?
Your turn1 of 2
Let A = [[3, 2], [7, -4]] and B = [[-7, -5], [3, -8]]. Find the (1,1) entry of AB.

The Transpose of a Matrix

  1. How it worksHow does taking the transpose turn every row of a matrix into a column?
  2. Worked exampleFind the transpose of the matrix [[1, 2, 3], [4, 5, 6]].
  3. Common mistakeWhy isn't the transpose of a product AB just the product of the transposes in the same order?
  4. Going furtherWhat happens to a symmetric matrix when you take its transpose?
Your turn1 of 2
Find the (3,2) entry of the transpose of the matrix [[4, 3, 8], [-4, -8, -5]].

Diagonal and Identity Matrices

  1. How it worksHow does multiplying by a diagonal matrix just scale each coordinate separately, with no mixing between them?
  2. Worked exampleFind the product of the diagonal matrix [[2, 0], [0, 3]] and the vector (5, 4).
  3. Common mistakeWhy isn't the identity matrix the only matrix that leaves every vector completely unchanged?
Your turn1 of 2
Find the first entry of the product of the diagonal matrix [[4, 0], [0, 3]] and the vector (8, -4).

The Trace of a Matrix

  1. How it worksHow does the trace of a matrix always turn out to equal the sum of its eigenvalues?
  2. Worked exampleFind the trace of the matrix [[4, 7], [2, 3]].
  3. Ties togetherHow do the trace and determinant of a 2 by 2 matrix together reveal its eigenvalues without factoring anything?
Your turn1 of 2
Find the trace of the matrix [[4, 3], [8, -4]].

Elementary Matrices

  1. How it worksHow does multiplying by an elementary matrix reproduce exactly one step of elimination?
  2. Worked exampleFind the elementary matrix that swaps row 1 and row 2 of a 3 by 3 matrix.
  3. Common mistakeWhy doesn't multiplying elementary matrices together in a different order reproduce the same overall row reduction?
  4. Ties togetherHow does chaining several elementary matrices together reproduce all of Gaussian elimination at once?
Your turn1 of 2
The elementary matrix E subtracts 5 times row 1 from row 2 (leaving row 3 unchanged). Applied to the vector (3, 8, -4), find the resulting second entry.

The Inverse of a Matrix

  1. How it worksHow does multiplying A by its inverse always collapse the product back down to the identity matrix?
  2. Worked exampleFind the inverse of the matrix [[2, 1], [1, 1]].
  3. Going furtherWhat happens to the inverse of a matrix when you swap two of its rows before inverting?
  4. Common mistakeWhy doesn't every square matrix have an inverse?
Your turn1 of 2
Find the (1,1) entry of the inverse of the matrix [[7, 2], [3, 5]].

When a Matrix Has No Inverse

  1. How it worksHow does elimination hitting a zero row reveal that a matrix is singular?
  2. Worked exampleShow that the matrix [[1, 2], [2, 4]] has no inverse.
  3. Common mistakeWhy doesn't a matrix full of nonzero numbers guarantee it has an inverse?
  4. Going furtherWhat happens when elimination is pushed all the way through on a matrix that turns out to be singular?
Your turn1 of 2
A 2 by 2 matrix has one row that is exactly 4 times the other row. What does this guarantee about the matrix?

Computing an Inverse with Gauss-Jordan

  1. How it worksHow does row-reducing the block [A given I] all the way to [I given A inverse] actually work?
  2. Worked exampleFind the inverse of the matrix [[1, 2], [3, 5]] using Gauss-Jordan elimination.
  3. Going furtherWhat happens during Gauss-Jordan elimination when the matrix on the left turns out to be singular?
Your turn1 of 2
Find the (2,2) entry of the inverse of the matrix [[7, 2], [3, 5]].

The Inverse of a Product and a Transpose

  1. How it worksHow does the inverse of AB turn out to be B inverse times A inverse, not the other way around?
  2. Worked exampleFind the inverse of AB given that A inverse is [[1, 0], [-1, 1]] and B inverse is [[2, 0], [0, 1]].
  3. Common mistakeWhy isn't the inverse of AB simply A inverse times B inverse?
Your turn1 of 2
Given that A inverse is [[3, 2], [7, -4]] and B inverse is [[-7, -5], [3, -8]], find the (1,1) entry of the inverse of AB.

LU Factorization

  1. How it worksHow does elimination on A split it into a lower triangular L and an upper triangular U?
  2. Worked exampleFind the LU factorization of the matrix [[2, 1], [4, 5]].
  3. Going furtherWhat happens to LU factorization when a row swap is needed partway through elimination?
  4. Ties togetherHow does having A equal LU make solving Ax = b faster when you need to solve it for many different b's?
Your turn1 of 2
The matrix [[4, 2], [7, -4]] factors as LU, where L = [[1, 0], [7/4, 1]]. Find the (2,2) entry of U.

Permutation Matrices and PLU

  1. How it worksHow does a permutation matrix reorder the rows of a matrix without changing any of its numbers?
  2. Worked exampleFind the permutation matrix that swaps rows 1 and 3 of a 3 by 3 matrix.
  3. Common mistakeWhy isn't a permutation matrix's inverse computed the same laborious way a general matrix's inverse is?
Your turn1 of 2
The permutation matrix that swaps rows 1 and 3 of a 3 by 3 matrix is applied to the vector (4, 3, 8). Find the resulting first entry.

Block Matrix Multiplication

  1. How it worksHow does splitting two matrices into matching blocks let you multiply them block by block?
  2. Worked exampleFind the product of two 4 by 4 matrices split into four 2 by 2 blocks each, where every off-diagonal block is zero.
  3. Common mistakeWhy doesn't block matrix multiplication work if the blocks aren't lined up in matching row-column order?
Your turn1 of 2
Two 4 by 4 matrices are each split into 2 by 2 blocks, with every off-diagonal block equal to zero. Their top-left blocks are [[3, 2], [7, -4]] and [[-7, -5], [3, -8]]. Find the (1,1) entry of the full product.