LINEAR ALGEBRA - STUDY GUIDE, CH 1-2
1. Is it possible for a linear system to have exactly two solutions? Explain.
2. Solve the following systems of equations using your calculator:
[several systems with two to four equations in three to four unknowns]
(For systems with infinite solutions, be able to write the parameterized solution)
3. Which of the systems in question 2 are
a. Homogeneous? (be able to give both trivial and nontrivial solutions)
b. Consistent?
c. Dependent?
4. Which of the systems in question 2 could be solved using
a. Gaussian elimination?
b. Gauss- Jordan elimination?
c. The inverse of the coefficient matrix?
d. Cramer’s rule?
5. Given the matrix A [3x3 matrix] find the a. Inverse, b. Transpose, c. Determinant, d. Trace, e. A-2
6. Given two matrices A and B of appropriate dimensions, find
a. A linear combination such as 3A+2B
b. The product AB
7. Perform the next two
(no more) elementary row operations in the Gauss-Jordan elimination process, as
explained in your textbook for this matrix [a 3x4 matrix which is
already partway through the Gauss-Jordan process]
8. Find the inverse of a 2x2 matrix using 
(This is the same as
the formula on page 44)
Do the appropriate multiplication using the resulting A-1 to solve a system of two equations.
9. Find a positive or negative power of A (e.g. A-1 or A3 or A-3) by inspection if A is a diagonal matrix.
10. Given a diagonal matrix D and another matrix A, calculate the products DA or AD without going through the full-blown matrix multiplication process -- you must know the shortcut.
11. Given a triangular matrix
a. Is it upper triangular or lower triangular? (Know the vocabulary)
b. Determine by inspection whether it is invertible or not; explain your reasoning.
12. Be able to do short proofs and solve short matrix equations, such as
a. Solve Ax=b for x, or prove that the solution is x=A-1b (Provided that A is invertible)
b. Prove that if A is invertible and A2 = I, then A = A-1.
c. Prove that if A is invertible and A2 = A, then A = I
d. Prove that the solution to D = Ax - x (a formula from Finite Math) is x = (A - I)-1 D.
e. Prove that if A, B and AB are invertible, then (AB)-1 = B-1A-1
f. Given that (AB)T = BTAT, prove that if A is invertible, then (A-1)T = (AT)-1
g. Given that det(AB)=det(A)det(B), prove that if A is invertible, det(A-1)=1/det(A)
13. Calculate a determinant using
a. Signed elementary products, and/or
b. Row reduction, and/or
c. Cofactor expansion.
14. Determine whether a given elementary product has an even or odd permutation.
15. Given a 2x2 matrix, find
a. Characteristic equation det(lI-A)=0
b. Eigenvalues (the solutions l)
c. Eigenvectors (solutions x to the equation (lI-A)x = 0)
16. Know and apply the theorems on exponents, transposes and inverses on pages 44-48.
17. Given a singular matrix, use various techniques (other than attempting to calculate the inverse) to show that it is not invertible. You must be able to relate this to the determinant, to the reduced row echelon form of the matrix, and to visual clues (i.e. a row or column of zeros, two equal rows or equal columns, or two proportional rows or proportional columns).
18. Solve a system of equations by Cramer’s Rule. Expect a system that you can’t put into your calculator, like
fx + ty = i
ox + Jy = v
Return to: Merced College; Don Power Updated 02/14/07 by Don Power