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When determining if two matrices, [tex]\( A \)[/tex] and [tex]\( B \)[/tex], are equal, we need to examine a few specific conditions. Let's evaluate each statement one by one:
1. Matrix [tex]\( A \)[/tex] must be a diagonal matrix.
This statement is false. Matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] do not need to be diagonal to be equal. Equality between two matrices depends only on the equality of their corresponding elements, not on their structure. For instance, two equal [tex]\( 2 \times 2 \)[/tex] matrices could be:
[tex]\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \][/tex]
Neither of these need to be diagonal matrices.
2. Both matrices must be square.
This statement is false. While many problems involve square matrices, matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] simply need to have the same dimensions. They could be rectangular, as long as both matrices share the same number of rows and columns. For example, two equal [tex]\( 2 \times 3 \)[/tex] matrices could be:
[tex]\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \][/tex]
3. Both matrices must be the same size.
This statement is true. For two matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] to be considered equal, they must have the same dimensions, meaning the same number of rows and columns. This is a fundamental requirement for matrix equality.
4. For any value of [tex]\( i, j \)[/tex], [tex]\( a_{ij} = b_{ij} \)[/tex].
This statement is true. If two matrices are equal, then every element at position [tex]\((i, j)\)[/tex] in matrix [tex]\( A \)[/tex] must be equal to the corresponding element at the same position in matrix [tex]\( B \)[/tex]. Mathematically, this is written as:
[tex]\[ A = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{pmatrix} = B = \begin{pmatrix} b_{11} & b_{12} & \dots & b_{1n} \\ b_{21} & b_{22} & \dots & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ b_{m1} & b_{m2} & \dots & b_{mn} \end{pmatrix} \][/tex]
This equality implies [tex]\( a_{ij} = b_{ij} \)[/tex] for all [tex]\( i \)[/tex] and [tex]\( j \)[/tex].
So, summarizing the statements:
- Matrix [tex]\( A \)[/tex] must be a diagonal matrix. (False)
- Both matrices must be square. (False)
- Both matrices must be the same size. (True)
- For any value of \( i, j: a_{ij} = b_{ij}. (True)
1. Matrix [tex]\( A \)[/tex] must be a diagonal matrix.
This statement is false. Matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] do not need to be diagonal to be equal. Equality between two matrices depends only on the equality of their corresponding elements, not on their structure. For instance, two equal [tex]\( 2 \times 2 \)[/tex] matrices could be:
[tex]\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \][/tex]
Neither of these need to be diagonal matrices.
2. Both matrices must be square.
This statement is false. While many problems involve square matrices, matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] simply need to have the same dimensions. They could be rectangular, as long as both matrices share the same number of rows and columns. For example, two equal [tex]\( 2 \times 3 \)[/tex] matrices could be:
[tex]\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \][/tex]
3. Both matrices must be the same size.
This statement is true. For two matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] to be considered equal, they must have the same dimensions, meaning the same number of rows and columns. This is a fundamental requirement for matrix equality.
4. For any value of [tex]\( i, j \)[/tex], [tex]\( a_{ij} = b_{ij} \)[/tex].
This statement is true. If two matrices are equal, then every element at position [tex]\((i, j)\)[/tex] in matrix [tex]\( A \)[/tex] must be equal to the corresponding element at the same position in matrix [tex]\( B \)[/tex]. Mathematically, this is written as:
[tex]\[ A = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{pmatrix} = B = \begin{pmatrix} b_{11} & b_{12} & \dots & b_{1n} \\ b_{21} & b_{22} & \dots & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ b_{m1} & b_{m2} & \dots & b_{mn} \end{pmatrix} \][/tex]
This equality implies [tex]\( a_{ij} = b_{ij} \)[/tex] for all [tex]\( i \)[/tex] and [tex]\( j \)[/tex].
So, summarizing the statements:
- Matrix [tex]\( A \)[/tex] must be a diagonal matrix. (False)
- Both matrices must be square. (False)
- Both matrices must be the same size. (True)
- For any value of \( i, j: a_{ij} = b_{ij}. (True)
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