Review

Dimensions

Name the dimensions of each matrix.

#1) A

#2) C

Entries

Using A and B, find the values of each entry.

#3)  a₁₁ =

#4)  b₁₁ =

Equal Matrices

#5)  Find the values of x and y.

#6)  Find the values of x, y and z.

Matrix Addition

Given the matrices A, B, C, D and E find each of the following sums and differences.

#7) A + B + C
#8) A - B - C
#9) A + B - C
#10) What must be true about 2 matrices in order to add them?

Scalar Multiplication

Find each product

#1) -2A

Matrix Multiplication?

Determine which products can be found. Justify your answer. If the product can be found, state the dimension of the matrix product.

#1) AB
#2) BA

Matrix Multiplication

Find each product given the matrices.

#1) AB
#2) AE

Calculations With Transposed Matrices

Given each matrix, find each calculation.

#7) F^T E +BC 

#8) H^TD + E

Determinants

Given the matrices, find each calculation.

#10) -|D| + 9|E|
#11) 10|H| + 5|G| - 2|E|

Are They Inverses?

Determine if each pair of matrices are inverses.


#2) Are C and D inverses?






#3) Are E and F inverses?

Multiplicative Inverses

Find the multiplicative inverse of each matrix. If not possible, write singular.

#5) E^-1
#6) F^-1

Solving Equations with Matrices

Solve each system of equations by using matrix equations.
#1)  −5x − 5y = 25
       −2x − 4y = 16


#2)  −x + 4y = −2
       −2x + 5y = −4

Solving a System Using Gaussian Elimination 

Write the system of equations in an augmented matrix.  Use elementary row operations to get the matrix into row-echelon form or reduced-row echelon form to solve the system.


#1)  3x − 4y = 1
       −5x + 2y = 3

#2)  −6x − 2y − z = −17
       5x + y − 6z = 19
       −4x − 6y − 6z = −20

Answers Dimensions

#1)  A_1x5 

#2)  C_8x1 

Answers Entries

#3)  a₁₁ = 52

#4)  b₁₁ = 1

Answers Solving For Variables

#5) (-2, -1), (2, -1)
#6) (18, -37, 19)

Answers Matrix Addition

#10) The 2 matrices have to have the same dimensions.

Answers Scalar Multiplication

Answers Matrix Multiplication?

#1) Yes because A is 3x2 and B is 2x4.  Since the 2's are the same, they can be multiplied with the resultant matrix being 3x4.

#2) No because B is 2x4 and A is 3x2.  Since the 4 and 3 aren't the same, they can NOT be multiplied.

Answers Matrix Multiplication

Answers Transposing

Answers Determinants
#10) -|D| = -179
#11) 10|H| + 5|G| - 2|E| = -163

Answers Are They Inverses?

#2) No
#3) Yes

Answers Multiplicative Inverses

Solving Equations with Matrices

#1) (−2, −3)
#2) (2, 0)

 Answers Solving a System Using Gaussian Elimination 

#1) (−1, −1)

#2) (2, 3, −1)