Matrix Multiplication Problems
- Posted by Brian Stocker
- Date June 17, 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 = a11 detA11 – a12 detA12 + a13 detA13 – … + (- 1)1+n a1n
detA1n
n
= Σ(- 1)1+j a1j detA1j
j=1
Here, A1j 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 A1j 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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