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Grade 12 · unit review

Review: Precalculus: Matrices and Linear Systems

Matrices organize linear operations and systems through rows and columns. Multiplication uses row-column products, a determinant tests two-dimensional invertibility, and valid row operations preserve a system’s solution set when applied to every entry.

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Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. Multiply [[1,2],[0,1]] by column [7,8]. What is the first component?
Need a hint?
  • A matrix acts on an input column by taking each row’s dot product with that column.
  • Row and column dimensions must be compatible.
2. Find the determinant of [[7,8],[8,11]].
Need a hint?
  • For a two-by-two matrix, determinant is the upper-left times lower-right minus the upper-right times lower-left.
  • Keep the subtraction order fixed; reversing it changes the determinant sign.
3. Which matrix is the inverse of [[1,7],[0,1]]?
Need a hint?
  • An inverse matrix undoes a matrix’s linear operation.
  • The inverse is not formed by reciprocating every entry.
4. In an augmented system, R1=[1,1|7], R2=[2,3|8]. Replace R2 with R2−2R1.
Need a hint?
  • Subtracting a multiple of one equation from another corresponds to subtracting that multiple of an augmented row.
  • Leaving the augmented value unchanged changes the system.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Multiply [[1,2],[0,1]] by column [9,10]. What is the first component? Candidate answers: (A) 28 (B) 29 (C) 30 (D) 58. 2. Find the determinant of [[9,10],[10,13]]. Candidate answers: (A) 16 (B) 17 (C) 18 (D) 34. 3. Which matrix is the inverse of [[1,9],[0,1]]? Candidate answers: (A) [[0,1],[1,9]] (B) [[1,0],[−9,1]] (C) [[1,9],[0,1]] (D) [[1,−9],[0,1]]. 4. In an augmented system, R1=[1,1|9], R2=[2,3|10]. Replace R2 with R2−2R1. Candidate answers: (A) R2=[0,1|-8] (B) R2=[0,1|1] (C) R2=[0,5|28] (D) R2=[1,2|1].

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 29. The first output is the first row dot the column: 1(9)+2(10)=29. The second output is 10; entries are not multiplied only by position. 2. 17. Determinant=upper-left×lower-right−upper-right×lower-left=9×13−10²=17. Its nonzero value means this 2×2 matrix is invertible. 3. [[1,−9],[0,1]]. Multiply [[1,9],[0,1]] by [[1,−9],[0,1]]. The off-diagonal sum −9+9=0 and diagonal entries remain 1, giving the identity matrix. 4. R2=[0,1|-8]. Subtract twice every entry of R1, including its augmented value: [2−2,3−2|10−2(9)]=[0,1|-8]. Such a row operation preserves the system’s solution set.

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