High School
SAT
ACT
University
MCAT
LSAT
DAT

Determinant Properties Practice Question

Related Topics

Wize University Linear Algebra Textbook > Determinants

Basics of Determinants

4 Activities

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
If 𝑨 = 𝑪-1𝟏𝑩𝑪, where C is an invertible matrix, and det(𝑩)=12;det(𝑪)=3.Calculate  det(𝑨).det(𝑩) = 12; det(𝑪) = 3. \text{Calculate}\ \ det(𝑨).




More Basics of Determinants Questions:
133 - FML 3 - 18.1W e.g. 17.3
If   ⁣A‾\bcb{\boldsymbol{ \ul{A} }} is a 3×3\bcb{\boldsymbol{ 3 \times 3}} matrix with det(  ⁣A‾)=−4\bcb{\boldsymbol{ \det{\ul{A}} = -4}}, find:
det(−AT)\boldsymbol{ \det{-\mtran{A}} }
133 - FML 3 - 18.1W e.g. 11
Find all possible values of det(  ⁣A‾)\bcb{\boldsymbol{ \det{\ul{A}} }} given that adj(A)=  [124−122018]\bcb{\boldsymbol{ adj(A) =  \begin{bmatrix} 1 & 2 & 4 \\ -1 & 2 & 2 \\ 0 & 1 & 8 \end{bmatrix} }}
133 - FML 3 - 18.1W e.g. 71
  ⁣B‾=  [1234−112x2]\bcb{\boldsymbol{ \ul{B} =  \begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 1 \\ 2 & x & 2 \end{bmatrix} }} such that det(  ⁣B‾)=−8 \bcb{\boldsymbol{ \det {\ul{B}} = -8 }} , then x\bcb{\boldsymbol{ x }} is:
(a)  111(b)  −1(c)  0(d)  None of the above\text{(a)}\;\boldsymbol{ \frac{1}{11} } \qquad\qquad\qquad \text{(b)}\; \boldsymbol{ -1}\qquad\qquad\qquad \text{(c)}\; \boldsymbol{ 0}\qquad\qquad\qquad \text{(d)}\; \text{\bf{None of the above}}
Basics of Determinants
Answer the following true/false questions.
Compute the determinant of
[31−111023−20341022]\begin{bmatrix} 3&1&−1&1\\ 1&0&2&3\\ −2&0&3&4\\ 1&0&2&2 \end{bmatrix}
Find the determinant of A=[0203−140−13110102−2]A=\begin{bmatrix} 0&2&0&3\\ −1&4&0&−1\\ 3&1&1&0\\ 1&0&2&−2 \end{bmatrix}
Basics of Determinants
Let CC be an invertible matrix with det(C)=3\det C = 3, and let B=[−2−1001000−5]B= \left[ \begin{array}{rrr} -2 & -1 & 0\\ 0 & 1 & 0\\ 0 & 0 & -5\\ \end{array} \right].
If A=C−1BCA = C^{-1}BC, calculate det(A)\det A.
Determinant Properties Practice Question
If 𝑨 = 𝑪-1𝟏𝑩𝑪, where C is an invertible matrix, and det(𝑩)=12;det(𝑪)=3.Calculate  det(𝑨).det(𝑩) = 12; det(𝑪) = 3. \text{Calculate}\ \ det(𝑨).
133 - FML 1 - 18.1W e.g. 7
Recall that diagonal matrices are substantially easier to work with than non-diagonal matrices:
e.g. If
A=diag(a,b,c)=[a000b000c]\text{A}=\text{diag}\left(a,b,c\right) = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix}
Practice: Determinant Property
Practice Question: Determinant Property
Let 𝐶 be an invertible matrix, det (𝐵) = 10, and det (𝐶) = 3.
If 𝐴 = 𝐶−1𝐵𝐶, calculate det (𝐴).
Basics of Determinants
Find det[32−4101−12−1−750−6−48−2]\text{det}\begin{bmatrix} 3&2&-4&1\\ 0&1&-1&2\\ -1&-7&5&0\\ -6&-4&8&-2 \end{bmatrix}.
Determinants: Basics
If A=[1234021−500−30000−1]A= \begin{bmatrix} 1&2&3&4\\ 0&2&1&-5\\ 0&0&-3&0\\ 0&0&0&-1 \end{bmatrix}, calculate det(Adj A)det(Adj\ A).
Determinants: Basics
AA is a 3×33\times 3 matrix where det A=2det\ A=2.
Find det(5A2A−1AT)det(5A2A^{-1}A^T).
Determinants: Basics
A=[45−12], B=[−10002100302010013−4−10], C=[−155−25], I10=10×10  identity matrixA= \begin{bmatrix} 4&5\\ -1&2 \end{bmatrix}, \ B= \begin{bmatrix} -1&0&0&0\\ 2&1&0&0\\ 30&20&10&0\\ 1&3&-4&-10 \end{bmatrix}, \ C= \begin{bmatrix} -1&5\\ 5&-25 \end{bmatrix}, \ I_{10}=10\times 10\ \text{ identity matrix}
Compute det(AC)−det(10BT)+det(2I10)det(AC)-det(10B^T)+det(2I_{10}).
Determinants: Basics
Find the determinant of the matrix [2−41−3]\begin{bmatrix}2&-4\\ 1&-3\end{bmatrix}.