If `[(1,4),(2,0)]`=`[(x,y^(2)),(z,0)]` `y lt 0` then x-y+z is equal to

Options

A. 5
B. 2
C. 1
D. `-3`

Answer

To solve the problem, we need to find the value of the expression \( x - y + z \) given the matrices:

\[ \begin{pmatrix} 1 & 4 \\ 2 & 0 \end{pmatrix} = \begin{pmatrix} x & y^2 \\ z & 0 \end{pmatrix} \]

### Step 1: Equate the corresponding elements of the matrices

From the equality of the matrices, we can equate the corresponding elements:

- \( a_{11} = b_{11} \) gives us \( 1 = x \)
- \( a_{12} = b_{12} \) gives us \( 4 = y^2 \)
- \( a_{21} = b_{21} \) gives us \( 2 = z \)
- \( a_{22} = b_{22} \) gives us \( 0 = 0 \) (which is trivially true)

### Step 2: Solve for \( x \), \( y \), and \( z \)

From the equations we derived:

1. From \( 1 = x \), we find:
\[ x = 1 \]

2. From \( 4 = y^2 \), we can solve for \( y \):
\[ y^2 = 4 \implies y = 2 \text{ or } y = -2 \]
However, we are given the condition \( y < 0 \), so we choose:
\[ y = -2 \]

3. From \( 2 = z \), we find:
\[ z = 2 \]

### Step 3: Substitute values into the expression \( x - y + z \)

Now we substitute the values of \( x \), \( y \), and \( z \) into the expression \( x - y + z \):

\[ x - y + z = 1 - (-2) + 2 \]

### Step 4: Simplify the expression

Now we simplify the expression:

\[ 1 + 2 + 2 = 5 \]

### Final Answer

Thus, the value of \( x - y + z \) is:

\[ \boxed{5} \]

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