Linear algebra practice exam with answers

A full practice exam covering Gaussian elimination, determinants, rank, inverse matrices and solvability. Answer online and get it checked at once, or print it with or without worked answers. New exams with different numbers come with a free account.

4 questions · 28 points · about 45 minutes

1. Question 1

   8 points

   How many solutions does the system have?

   Bring the augmented matrix to row echelon form and compare rank(A) with rank(A|b).

   IIIIII

   2	−1	1	−4
   0	2	3	1
   −2	−1	−4	3

   Apply

   IIIIII↔−+·/⌫

   Hint: next step

   Your answer

   Exactly one solutionNo solutionInfinitely many solutions

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2. Question 2

   8 points

   Find the rank of the matrix.

   Bring the matrix to row echelon form and count the rows that are not all zeros.

   IIIIIIIV

   1	−2	3	1
   1	−2	3	5
   −2	4	−6	−4
   1	−2	3	3

   Apply

   IIIIIIIV↔−+·/⌫

   Hint: next step

   Your answer

   Rank =Check

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3. Question 3

   8 points

   Solve the system of linear equations.

   Use row operations until you can read off the solution, then enter it below.

   IIIIIIIV

   −4	−8	12	2	−18
   −4	−7	6	−4	−1
   −2	−5	4	−4	1
   −2	−3	2	−2	1

   x₁, x₂, x₃, x₄ | b

   Apply

   IIIIIIIV↔−+·/⌫

   Hint: next step

   Your answer

   x₁ =x₂ =x₃ =x₄ =Check

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4. Question 4

   4 points

   Calculate the determinant.

   Swapping rows flips the sign; multiplying a row by k multiplies the determinant by k. Adding a multiple of another row changes nothing.

   IIIIII

   −2	1	2
   2	1	2
   4	0	2

   Apply

   IIIIII↔−+·/⌫

   Hint: next step

   Your answer

   det =Check

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