Matrix Multiplication Problems

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 = a₁₁ detA₁₁ – a₁₂ detA₁₂ + a₁₃ detA₁₃ – … + (- 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.

Find the determinant of matrix A

a. – 30
b. – 25
c. 10
d. 15

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