Common Core · High school
HSN-VM.C.8
High school — Add, subtract and multiply matrices of appropriate dimensions.
A note on this mapping: HSN-VM.C.8 is a (+) standard — beyond the college- and career-ready threshold, i.e. precalculus rather than Algebra II. The transpose and the symmetric matrix are named nowhere in the Common Core; they arrive with the university course.
The 3 skills that cover it
- Matrix OperationsArithmetic on rectangles of numbers: add and scale matrices, multiply them row by column, work out which products are even defined and what size they come out, take a transpose, recognise a symmetric matrix and the identity, and see for yourself that AB and BA are usually different.Read the lesson →
- Linear TransformationsA function that moves every vector at once, and the matrix that describes it: test the two rules a linear map obeys, build the standard matrix from the images of the basis vectors, recognise rotations, reflections, scalings and shears, and read a composition as a matrix product.Read the lesson →
- Diagonalization & Its UsesChoosing the basis that makes a matrix simple: decide whether a matrix can be diagonalised, write down the diagonal matrix of eigenvalues, raise a matrix to a high power through its eigenbasis, and read the long run of a two-state chain or a population model off the eigenvalues.Read the lesson →
What a student should be able to do
- Add matrices, multiply one by a scalar, and take a transpose.
- Multiply two matrices, and give a single entry of a product without computing the rest.
- Decide whether a product is defined and what size it is.
- Recognise the identity and a symmetric matrix, and show that matrix multiplication is not commutative.
- Decide whether a map is linear, and name the rule that fails when it is not.
- Find the image of a vector under a transformation given by a matrix.
- Build the standard matrix of a transformation from the images of the basis vectors.
- Recognise the matrix of a rotation, reflection, scaling or shear, and compose two transformations by multiplying their matrices.
- Decide whether a matrix is diagonalisable, and say why when it is not.
- Give the diagonal matrix D in A = PDP⁻¹, and an entry of a matrix power computed through it.
- Find the steady-state vector of a two-state chain.
- Say which eigenvalue governs the long-run behaviour of a model, and what the eigenvalues mean in it.