Common Core · High school
HSA-REI.C.9
High school — Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
A note on this mapping: HSA-REI.C.9 is a (+) standard — beyond the college- and career-ready threshold, i.e. precalculus rather than Algebra II. The Common Core hands a 3 × 3 to technology. Row reduction is the procedure behind that button, and it is nobody's standard on its own.
The 4 skills that cover it
- Row Reduction & Gaussian EliminationThe one procedure the whole subject leans on: apply a row operation and say what it did, clear a column below a pivot, drive a matrix all the way to reduced row echelon form, count the pivots, and recognise when a matrix is not in that form yet and which rule it breaks.Read the lesson →
- Linear Systems in Matrix FormA system of equations rewritten as one matrix equation: build the augmented matrix, read a matrix back as a system, solve two and three unknowns by elimination, and decide from the reduced form whether there is one solution, none at all, or a whole line of them.Read the lesson →
- Inverse MatricesThe matrix that undoes another: invert a 2 by 2 with the formula, decide from a determinant whether an inverse exists at all, find one by reducing A beside the identity, solve a system with it, and collect the many equivalent ways of saying a matrix is invertible.Read the lesson →
- Rank & the Shape of SolutionsCounting what a matrix really pins down: read the rank off a reduced form, work out the nullity from it, count free variables, compare the rank of a matrix with the rank of its augmented form to decide consistency, and say the largest rank a given shape allows.Read the lesson →
What a student should be able to do
- Apply a row operation to a matrix and give the resulting entry, or name the operation that turned one matrix into another.
- Find the multiplier that clears an entry below a pivot, and carry out forward elimination.
- Reduce a matrix to reduced row echelon form.
- Count the pivots of a matrix and decide whether it is already in reduced row echelon form.
- Write a system as an augmented matrix, and read an augmented matrix back as a system.
- Solve a system of two or three equations by row reduction.
- Decide from a reduced matrix whether a system has one solution, no solution, or infinitely many.
- Describe an infinite solution set with a free variable, and find the value that makes a system inconsistent.
- Find the inverse of a 2 by 2 matrix, and decide from the determinant whether a matrix has one.
- Find an entry of an inverse by reducing the matrix beside the identity.
- Solve a matrix equation using an inverse.
- Apply the rule for the inverse of a product, and recognise equivalent statements of invertibility.
- Find the rank of a matrix from its reduced row echelon form, and the number of free variables.
- Use the rule nullity = n - rank to find one of the three from the other two.
- Compare the rank of a coefficient matrix with the rank of the augmented matrix to decide consistency.
- Give the largest possible rank of a matrix of a stated shape, and what row operations do to rank.