Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 150 minutes, as an unvalidated planning estimate. This collection contains 3 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Attempt the three original public problems before revealing solutions. Give full proof dependencies and state the field for each calculation.
Each rubric assesses a mathematical argument rather than draft length. Saved public practice is not an independently graded or secure examination.
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Over an arbitrary field, prove det(Iₙ+tuvᵀ)=1+t vᵀu directly by column multilinearity and alternation, for n≥1. Do not assume the matrix is invertible. Apply it in dimension three to u=(1,2,1) and v=(1,−1,3), with integer entries interpreted in the stated field. Classify the exceptional parameter over fields of characteristic two and fields of other characteristics. At an exceptional parameter, determine the kernel, rank and exact consistency condition for (I+tuvᵀ)x=b. Explain why the determinant calculation alone did not give that condition.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Over a field consider A(t) with rows (t,1,0),(0,t,0),(0,0,1), and D(t)=diag(t,t,1). Compute both determinants and adjugates, checking the product identities. At t=0 compare the ranks of each matrix and each adjugate. Then prove the general statement for an n×n matrix with n≥2: rank at most n−2 implies zero adjugate; rank n−1 implies adjugate of rank one. Use cofactors, the rank/minor criterion and the identity A adj(A)=0 when singular. Explain why this does not justify solving every singular system by dividing the adjugate identity.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let a four-by-four real matrix M have blocks P,Q above R,S, where P=(1,1;0,1), Q=I₂, R=diag(1,2), S=(3,1;1,4). Here semicolons separate rows. Compute det(M) by deriving a legal block elimination, then determine whether the same integer matrix is invertible after reduction to F2. Explain why a zero leading block of an unrelated block matrix would not imply singularity of the full matrix. Finally compare the real matrices D_s=diag(s,1/s), s≥1, which all have determinant one: give an exact right-hand-side perturbation proving that their worst relative solution amplification can grow as s².
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Over an arbitrary field, prove det(Iₙ+tuvᵀ)=1+t vᵀu directly by column multilinearity and alternation, for n≥1. Do not assume the matrix is invertible. Apply it in dimension three to u=(1,2,1) and v=(1,−1,3), with integer entries interpreted in the stated field. Classify the exceptional parameter over fields of characteristic two and fields of other characteristics. At an exceptional parameter, determine the kernel, rank and exact consistency condition for (I+tuvᵀ)x=b. Explain why the determinant calculation alone did not give that condition.
Over a field consider A(t) with rows (t,1,0),(0,t,0),(0,0,1), and D(t)=diag(t,t,1). Compute both determinants and adjugates, checking the product identities. At t=0 compare the ranks of each matrix and each adjugate. Then prove the general statement for an n×n matrix with n≥2: rank at most n−2 implies zero adjugate; rank n−1 implies adjugate of rank one. Use cofactors, the rank/minor criterion and the identity A adj(A)=0 when singular. Explain why this does not justify solving every singular system by dividing the adjugate identity.
Let a four-by-four real matrix M have blocks P,Q above R,S, where P=(1,1;0,1), Q=I₂, R=diag(1,2), S=(3,1;1,4). Here semicolons separate rows. Compute det(M) by deriving a legal block elimination, then determine whether the same integer matrix is invertible after reduction to F2. Explain why a zero leading block of an unrelated block matrix would not imply singularity of the full matrix. Finally compare the real matrices D_s=diag(s,1/s), s≥1, which all have determinant one: give an exact right-hand-side perturbation proving that their worst relative solution amplification can grow as s².
