Is the equation given below valid?
`[(2,3),(7,8),(9,1)]-[(2,3),(7,9),(9,1)]=[(1,4),(3,2)]-[(1,4),(3,2)]`

Answer

To determine if the equation

\[ [(2,3),(7,8),(9,1)] - [(2,3),(7,9),(9,1)] = [(1,4),(3,2)] - [(1,4),(3,2)] \]

is valid, we will analyze both sides of the equation step by step.

### Step 1: Identify the matrices
Let's denote the matrices as follows:
- Let \( A = \begin{pmatrix} 2 & 3 \\ 7 & 8 \\ 9 & 1 \end{pmatrix} \)
- Let \( B = \begin{pmatrix} 2 & 3 \\ 7 & 9 \\ 9 & 1 \end{pmatrix} \)
- Let \( C = \begin{pmatrix} 1 & 4 \\ 3 & 2 \end{pmatrix} \)
- Let \( D = \begin{pmatrix} 1 & 4 \\ 3 & 2 \end{pmatrix} \)

### Step 2: Determine the dimensions of the matrices
- Matrix \( A \) and matrix \( B \) are both \( 3 \times 2 \) matrices (3 rows and 2 columns).
- Matrix \( C \) and matrix \( D \) are both \( 2 \times 2 \) matrices (2 rows and 2 columns).

### Step 3: Perform the subtraction on the left-hand side
Now, we will compute the left-hand side:

\[ A - B = \begin{pmatrix} 2 & 3 \\ 7 & 8 \\ 9 & 1 \end{pmatrix} - \begin{pmatrix} 2 & 3 \\ 7 & 9 \\ 9 & 1 \end{pmatrix} \]

Calculating this gives:

\[ = \begin{pmatrix} 2-2 & 3-3 \\ 7-7 & 8-9 \\ 9-9 & 1-1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & -1 \\ 0 & 0 \end{pmatrix} \]

### Step 4: Perform the subtraction on the right-hand side
Now, we will compute the right-hand side:

\[ C - D = \begin{pmatrix} 1 & 4 \\ 3 & 2 \end{pmatrix} - \begin{pmatrix} 1 & 4 \\ 3 & 2 \end{pmatrix} \]

Calculating this gives:

\[ = \begin{pmatrix} 1-1 & 4-4 \\ 3-3 & 2-2 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

### Step 5: Compare the results
Now we compare the results of both sides:
- Left-hand side: \( \begin{pmatrix} 0 & 0 \\ 0 & -1 \\ 0 & 0 \end{pmatrix} \) (which is a \( 3 \times 2 \) matrix)
- Right-hand side: \( \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \) (which is a \( 2 \times 2 \) matrix)

### Conclusion
Since the left-hand side is a \( 3 \times 2 \) matrix and the right-hand side is a \( 2 \times 2 \) matrix, they cannot be equal. Therefore, the equation is **not valid**.

### Final Answer
The equation is **false**.

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