Linear Algebra Practice Problems

Linear algebra is the branch of mathematics that studies vectors, matrices, and linear equations. It provides the mathematical tools to solve systems of equations and describe relationships in many dimensions.

Solved Examples

Example 1: Find the sum of the two vectors \overrightarrow{\rm A} = 2i + 3j + 5k$ and \overrightarrow{\rm B} = -i + 2j + k$.

Solution:

\overrightarrow{\rm A} + \overrightarrow{\rm B} = (2-1)i + (2 + 3)j + (5 + 1)k = i + 5j + 6k

Example 2: Find the dot product of \overrightarrow{\rm P} = -2i + j + 3k and \overrightarrow{\rm Q}= i - 2j + k

Solution:

\overrightarrow{\rm P}.\overrightarrow{\rm Q} = -2i(i - 2j + k) + j(i - 2j + k) + 3k(i - 2j + k)

= -2i -2j + 3k

Example 3: Find the solution of x + 2y = 3 and 3x + y = 5

Solution:

From x + 2y = 3 we get x = 3 - 2y

Putting this value of x in the second equation we get

3(3 - 2y) + y = 5
⇒ 9 - 6y + y = 5
⇒ 9 - 5y = 5
⇒ -5y = -4
⇒ y = 4/5

Putting this value of y in 1st equation we get

x + 2(4/5) = 3
⇒ x = 3 - 8/5
⇒ x = 7/5

Example 4: Matrix Multiplication, Find the product of the matrices:

\:A=\left(\begin{matrix}\mathbf{1}&\mathbf{2}\\\mathbf{3}&\mathbf{4}\\\end{matrix}\right),\ \ \ \ B=\left(\begin{matrix}\mathbf{5}&\mathbf{6}\\\mathbf{7}&\mathbf{8}\\\end{matrix}\right)

Solution:

AB=\left(\begin{matrix}1\bullet5+2\bullet7&1\bullet6+2\bullet8\\3\bullet5+4\bullet7&3\bullet6+4\bullet8\\\end{matrix}\right)=\left(\begin{matrix}5+14&6+16\\15+28&18+32\\\end{matrix}\right)=\left(\begin{matrix}19&22\\43&50\\\end{matrix}\right)

Example 5: Eigenvalues of a Matrix. Find the eigenvalues of the matrix:

A=\left(\begin{matrix}\mathbf{3}&\mathbf{8}\\\mathbf{0}&\mathbf{6}\\\end{matrix}\right)

Solution:

1. Write the characteristic equation:

A\ -\ \lambda I

2. Find the determinant (det) of characteristic equation:

\left|A - \lambda I\right|=\left|\begin{matrix}\mathbf{3}-\lambda&\mathbf{8}\\\mathbf{0}&\mathbf{6}-\lambda\\\end{matrix}\right|=(\mathbf{3}-\lambda)(\mathbf{6}-\lambda)-\mathbf{8}\bullet\mathbf{0}=(\mathbf{3}-\lambda)(\mathbf{6}-\lambda)

3. Equate the determinant with Zero "0":

(\mathbf{3}-\lambda)(\mathbf{6}-\lambda)=0 \Rightarrow \lambda=3,6

Therefore, the eigenvalues are 3, 6.

Practice Problems

Question 1: Solve the system of equations:

• x + y + z = 6
• 2x + 3y + 5z = 4
• 4x + 3y + z = 2

Question 2: Find the eigenvalues and eigenvectors of the matrix:

\left(\begin{matrix}\mathbf{5}&\mathbf{0}\\\mathbf{7}&\mathbf{8}\\\end{matrix}\right)

Question 3: Find the determinant of the matrix:

\left(\begin{matrix}\mathbf{3}&\mathbf{6}\\\mathbf{4}&\mathbf{8}\\\end{matrix}\right)

Question 4: Find the product of the matrices:

A=\left(\begin{matrix}\mathbf{1}&\mathbf{2}\\\mathbf{6}&\mathbf{4}\\\end{matrix}\right),\ \ \ \ B=\left(\begin{matrix}\mathbf{5}&\mathbf{4}\\\mathbf{0}&\mathbf{2}\\\end{matrix}\right)