If [((x - y), 2y), ((2y + z), (x + y))] = [(1, 4), (9, 5)],\(\begin{bm

1.	If [((x - y), 2y), ((2y + z), (x + y))] = [(1, 4), (9, 5)],\(\begin{bmatrix}x-y&2y\\2y + z&x + y\end{bmatrix}\) = \(\begin{bmatrix}1&4\\9&5\end{bmatrix}\).then write the value of (x + y + z).
Answer» We have \(\begin{bmatrix}x-y&2y\\2y + z&x + y\end{bmatrix}\) = \(\begin{bmatrix}1&4\\9&5\end{bmatrix}\). We know that if two matrices are equal then their corresponding elements are also equal. Now, equating the corresponding elements of both matrices, we get 2y = 4 ⇒ y = 2. ... (1) (By equating a₁₂ elements of both matrices.) x − y = 1 ⇒ x = 1 + y ⇒ x = 1 + 2 ⇒ x = 3. ... (2) (By equating a₁₁ elements of both matrices and putting the value of y = 2 from equation (1).) 2y + z = 9 ⇒ z = 9 − 2y ⇒ z = 9 − 4 ⇒ z = 5. ... (3) (By equating a₂₁ elements of both matrices and putting the value of y = 2 from equation (1).) x + y = 5 ⇒ 3 + 2 = 5 ⇒ 5 = 5 (Satisfying) (By putting x = 3 and y = 2 from equations (2) and (1), respectively) (By equating a₂₂ elements of both matrices.) Therefore, x = 3, y = 2 and z = 5. Therefore, x + y + z = 3 + 2 + 5 = 10. Hence, if \(\begin{bmatrix}x-y&2y\\2y + z&x + y\end{bmatrix}\) = \(\begin{bmatrix}1&4\\9&5\end{bmatrix}\). Then, the value of (x + y + z) is 10.

If [((x - y), 2y), ((2y + z), (x + y))] = [(1, 4), (9, 5)],\(\begin{bmatrix}x-y&2y\\2y + z&x + y\end{bmatrix}\) = \(\begin{bmatrix}1&4\\9&5\end{bmatrix}\).then write the value of (x + y + z).

Answer»

We have \(\begin{bmatrix}x-y&2y\\2y + z&x + y\end{bmatrix}\) = \(\begin{bmatrix}1&4\\9&5\end{bmatrix}\).

We know that if two matrices are equal then their corresponding elements are also equal.

Now, equating the corresponding elements of both matrices, we get

2y = 4 ⇒ y = 2. ... (1) (By equating a₁₂ elements of both matrices.)

x − y = 1 ⇒ x = 1 + y ⇒ x = 1 + 2 ⇒ x = 3. ... (2)

(By equating a₁₁ elements of both matrices and putting the value of y = 2 from equation (1).)

2y + z = 9 ⇒ z = 9 − 2y ⇒ z = 9 − 4 ⇒ z = 5. ... (3)

(By equating a₂₁ elements of both matrices and putting the value of y = 2 from equation (1).)

x + y = 5 ⇒ 3 + 2 = 5 ⇒ 5 = 5 (Satisfying) (By putting x = 3 and y = 2 from equations (2) and (1), respectively)

(By equating a₂₂ elements of both matrices.)

Therefore, x = 3, y = 2 and z = 5.

Therefore, x + y + z = 3 + 2 + 5 = 10.

Hence, if \(\begin{bmatrix}x-y&2y\\2y + z&x + y\end{bmatrix}\) = \(\begin{bmatrix}1&4\\9&5\end{bmatrix}\). Then, the value of (x + y + z) is 10.