Properties of the Determinant
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- Determinant for Rows or Columns with Zero Let AAA be a square matrix with a least one row or column consisting of only zeroes. Then det A=0det\ A=0det A=0. Example A=[1200]det A…
By Alex Federspiel
- Determinants from Row Operations Row Switch Let AAA be a square matrix and let BBB be the new matrix created when two rows or columns have been switched. Then det B=−det Adet\ B…
By Alex Federspiel
Problems 7
- A=[20000400008000016]A = \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 16 \end{bmatrix}A=20000400008000016 What is det(A)\det(A)det(A)?
- Let AAA and BBB be n×nn \times nn×n matrices. Which of the following facts about the determinant are true?
- Let AAA be a 4×44 \times 44×4 matrix and suppose det(A)=3\det(A) = 3det(A)=3. What is det(2A)\det(2A)det(2A)?
- Let aaa, bbb, ccc, xxx, yyy, and zzz be real numbers that satisfy the three equations 13x+by+cz=0ax+23y+cz=0ax+by+42z=0.\begin{align*} 13x + by + cz &= 0 \\ ax + 23y + cz &= 0 \…
- Without calculating, find the determinant of [429182921312813141228121220132824132521152921]\begin{bmatrix}4&29&1&8&29\\2&13&1&28&13\\14&12&28&12&12\\20&13&28&24&13\\25&21&15&29…
- The determinant of matrix AAA is 10. Then the following row operations were made: R2⟷R63R3⟶R3R5⟷R11248R3+R5⟶R5R2⟷R1112 R8⟶R8\\\begin{align}R_2&\longleftrightarrow R_6\\ 3R_3&\lo…
- The determinant for a square matrix AAA is 000. What is the determinant of A−1A^{-1}A−1?