Find the value of a , b , c , d for the matrix equation a − b 2 a + c 2 a − b 3 c + d = 1 5 0 13

Given, a − b 2 a + c 2 a − b 3 c + d = 1 5 0 13 We know that two matrices are equal if and only if they are of the same order, and their corresponding elements are equal. So, we have a - b = 1 → i 2 a + c = 5 → i i 2 a - b = 0 → i i i 3 c + d = 13 → i v Now, applying 2 i - i i i 2 a - 2 b - 2 a - b = 2 - 0 ⇒ 2 a - 2 b - 2 a + b = 2 ⇒ b = - 2 → v Substituting the value of b in equation i , we have a - - 2 = 1 ⇒ a = 1 - 2 = - 1 → v i Now, substituting the value of a in equation i i , we have 2 - 1 + c = 5 ⇒ c = 5 + 2 = 7 → v i i Now, substituting the value of c in equation i v , we have 3 7 + d = 13 ⇒ d = 13 - 21 = - 8 → v i i i From v , v i , v i i , and v i i i The required values are a = - 1 , b = - 2 , c = 7 , and d = - 8

Find the values of x and y if 2 1 3 0 x + y 0 1 2 = 5 6 1 8 .

Given, 2 1 3 0 x + y 0 1 2 = 5 6 1 8 We know that in scalar multiplication, each entry in the matrix is multiplied by the given scalar. 2 6 0 2 x + y 0 1 2 = 5 6 1 8 ⇒ 2 + y 0 + 6 0 + 1 2 x + 2 = 5 6 1 8 ⇒ y + 2 6 1 2 x + 2 = 5 6 1 8 We know that two matrices are equal if and only if they are of the same order and their corresponding elements are equal. ⇒ y + 2 = 5 and 2 x + 2 = 8 ⇒ y = 5 - 2 = 3 and 2 x = 8 - 2 = 6 ⇒ x = 3 Hence, x = 3 and y = 3

Find the value of x and y if 3 x + y - y 2 y - x 3 = 1 2 - 5 3

Given 3 x + y - y 2 y - x 3 = 1 2 - 5 3 Comparing the corresponding elements of given matrix we get - y = 2 Therefore y = - 2 Again we have 3 x + y = 1 3 x = 1 - y Substituting the value of y we get 3 x = 1 - ( - 2 ) 3 x = 1 + 2 3 x = 3 x = 3 3 x = 1 Hence, the required value of x is 1 and y is - 2 .

If A = 1 2 3 4 , find A + A T .

Given, A = 1 2 3 4 We know that the transpose of a matrix is a new matrix that is obtained by exchanging the rows and columns of the original matrix. Thus, A T = 1 3 2 4 Now, A + A T = 1 2 3 4 + 1 3 2 4 = 1 + 1 2 + 3 3 + 2 4 + 4 = 2 5 5 8 Hence, A + A T = 2 5 5 8

If A = 1 2 3 , find A A T .

Given, A = 1 2 3 To find: A A T We know that the transpose of a matrix is a new matrix that is obtained by exchanging the rows and columns of the original matrix. Thus, A T = 1 2 3 Now, A A T = 1 2 3 1 2 3 = 1 × 1 + 2 × 2 + 3 × 3 = 14 Hence, A A T = 14