# Matrix Multiplication Problems

- Posted by Brian Stocker
- Date Published June 17, 2020
- Date modified July 8, 2020
- Comments 0 comment

### Operations with Matrices

Matrices are rectangular arrays, arranged in rows and columns. The dimension of an array is the number or rows times the number of columns.

Here, the dimension of the matrix below is 2 × 2

Two matrices with the same number of rows and columns can be added or subtracted element by element .

To multiply 2 matrices, the first matrix must have the same number of rows and the columns in the second.

**Finding the Determinate of a Matrix **

The determinant of a matrix is found by the formula:

det A = a_{11} detA_{11} – a_{12} detA_{12} + a_{13} detA_{13} – … + (- 1)1+n a_{1n}

detA_{1n}

n

= Σ(- 1)^{1+j} a_{1j} detA_{1j}

j=1

Here, A_{1j} named matrices are the submatrices obtained by

closing 1st row and column j in the matrix A. The closed row

and column elements are eliminated and the remaining entries

form A_{1j} submatrices. The determinant of a 2 x 2 matrix

is obtained by:

if A =

DET A = ab – cd.

### Multiplying Matrix

Find the determinant of matrix A

a. – 30

b. – 25

c. 10

d. 15

### Answer Key

**1.**

– (-1)

**2. D**

Notice that the dimensions for matrix A and B are 2 x 2

and 2 x 2, respectively. In matrix multiplication; the dimensions

are important. Since A * X = B, the dimensions

of matrix X should be 2 x 2. Let us say that matrix X is as

follows:

Now, let us write equations obtained from matrix multiplication:

1a + 4c = 2 … (I)

1b + 4d = 2 … (II)

1a + 3c = 5 … (III)

1b + 3d = 1 … (IV)

Now, we have 4 unknowns and 4 equations which means that we will be able to find the values of a, b, c and d. Using

equations (I) and (III), we will find a and c; using equations

(II) and (IV), we will find b and d:

1a + 4c = 2 … (I)

– / 1a + 3c = 5 … (III)

4c – 3c = 2 – 5

c = – 3

Inserting this value into equation (I): 1a + 4(- 3) = 2

a – 12 = 2

a = 14

1b + 4d = 2 … (II)

– / 1b + 3d = 1 … (IV)

4d – 3d = 2 – 1

d = 1

Inserting this value into equation (II): 1b + 4(1) = 2

b + 4 = 2

b = – 2

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