Subtract the Cross Products in a Two-by-Two Determinant
For a two-by-two matrix, determinant is the upper-left times lower-right minus the upper-right times lower-left.
About 25 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Find the determinant of [[2,3],[3,6]]. Candidate answers: (A) 2 (B) 3 (C) 4 (D) 6.
- Independent check inputs: Find the determinant of [[3,4],[4,7]]. Candidate answers: (A) 10 (B) 4 (C) 5 (D) 6.
Learn the idea
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. For a two-by-two matrix, determinant is the upper-left times lower-right minus the upper-right times lower-left. A nonzero determinant means the associated linear map has an inverse.
Here is the method on the supplied teaching case. Determinant=upper-left×lower-right−upper-right×lower-left=2×6−3²=3. Its nonzero value means this 2×2 matrix is invertible.
Keep the subtraction order fixed; reversing it changes the determinant sign. A determinant is a scalar, not a matrix, and zero signals a failure of this invertibility condition rather than an arbitrary division step.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Find the determinant of [[2,3],[3,6]]. Candidate answers: (A) 2 (B) 3 (C) 4 (D) 6. Solution: Determinant=upper-left×lower-right−upper-right×lower-left=2×6−3²=3. Its nonzero value means this 2×2 matrix is invertible. The final answer is 3. Keep the subtraction order fixed; reversing it changes the determinant sign. A determinant is a scalar, not a matrix, and zero signals a failure of this invertibility condition rather than an arbitrary division step.
Check your reasoning
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Helpful tips
- 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.
See the answer and explanation
5
Determinant=upper-left×lower-right−upper-right×lower-left=3×7−4²=5. Its nonzero value means this 2×2 matrix is invertible.
Take it further
Changed-input task: Find the determinant of [[9,10],[10,13]]. Candidate answers: (A) 16 (B) 17 (C) 18 (D) 34. Show the mathematical reason, not only the selected answer. Expected result: 17. Determinant=upper-left×lower-right−upper-right×lower-left=9×13−10²=17. Its nonzero value means this 2×2 matrix is invertible.
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