WEBVTT
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Given that π΄ is the matrix with first row three, three, three and second row three, three, three and π΅ is the matrix with first row three, three and second row three, three, is it true that the matrix π΄ is equal to the matrix π΅?
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Letβs start by recalling what we mean when we say that two matrices are equal.
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If we have the entry in row π and column π of matrix π΄ is π ππ and the entry in row π and column π of matrix π΅ is π ππ, then if π ππ is equal to π ππ for all of our values of π and π, we say that the matrix π΄ is equal to the matrix π΅.
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Otherwise, we say that these matrices are not equal.
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So to check the two matrices are equal, we need to check that all of their entries are identical.
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Letβs start with matrix π΄.
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We can see this has two rows and three columns.
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So in our case, what would our values of π ππ be?
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First, our matrix π΄ has two rows and three columns, so our values of π range from one to two and our values of π range from one to three.
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We can then do something similar for our matrix π΅.
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If π ππ is the entry in matrix π΅ in row π and column π, then because our matrix π΅ only has two rows and two columns, our values of π range from one to two and our values of π also range from one to two.
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But now we can start to see our problem from our definition.
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The entries must be equal for all possible rows and columns.
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Our matrix π΄ has three columns.
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However, our matrix π΅ only has two columns.
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So these matrices canβt possibly be equal.
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For example, if we highlight the two entries in matrix π΄ in column three, by our definition of equality, we would have to have a third column in matrix π΅ which is equal to this column.
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So in this case, the matrix π΄ is not equal to the matrix π΅.
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In fact, we can use exactly the same line of reasoning as we did in this question to deduce that if two matrices have different orders, they canβt be equal.
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In other words, if they have a different number of rows or columns, they canβt be equal.
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This means what weβve shown is for two matrices to be equal, they must have the same number of rows or columns.
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In other words, they must have the same order.