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State whether the following statement is true or false.
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If π΄ and π΅ are both two-by-two matrices, then π΄π΅ is never the same as π΅π΄.
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In order to prove that a statement is false, we simply need to find one example where the statement is not true.
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We are told that both of our matrices are two by two.
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And we will let the elements of matrix π΄ be π, π, π, π.
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Whilst we could let the elements of matrix π΅ have any values, in this case, we will let matrix π΅ be the identity matrix: one, zero, zero, one.
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We know that the identity matrix has ones on its leading diagonal and zeros everywhere else.
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To calculate matrix π΄π΅, we need to multiply π, π, π, π by one, zero, zero, one.
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When multiplying matrices, we multiply the elements of each row in the first matrix by each column in the second matrix.
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π multiplied by one is equal to π, and π multiplied by zero is zero.
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Therefore, the first element in the matrix π΄π΅ is π.
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Repeating this for the other rows and columns, we get the elements π, π, and π.
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Matrix π΄π΅ is equal to π, π, π, π, which is equal to matrix π΄.
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We will now repeat this method when multiplying matrix π΅, the identity matrix, by matrix π΄.
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Once again, this gives us the elements π, π, π, π.
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We have therefore found an example where the matrix π΄π΅ is the same as the matrix π΅π΄.
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This leads us to a general rule.
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When we multiply any matrix by the identity matrix, it is the same as multiplying the identity matrix by this matrix.
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In both cases, the original matrix remains the same.
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π΄πΌ is equal to πΌπ΄, which is equal to the matrix π΄.
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We can actually go one stage further when looking at the commutative property of matrices.
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We will now let matrix π΅ have the elements π, π, π, β.
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Multiplying the matrices π΄π΅ and π΅π΄, we get the following two-by-two matrices.
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At first glance, matrix π΄π΅ and π΅π΄ appear to have nothing in common.
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However, we do notice that the top-left element contains ππ or ππ and the bottom-right element contains πβ or βπ.
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The elements π, π, π, and β are the elements on the leading diagonals of matrices π΄ and π΅, respectively.
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We can see that if all the other products were equal to zero, the two matrices would be the same.
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Letβs consider what happens if π, π, π, and π are all equal to zero.
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Matrices π΄π΅ and π΅π΄ are both equal to ππ, zero, zero, πβ.
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This is an example of a diagonal matrix, as all the elements apart from those on the leading diagonal are equal to zero.
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This leads us to another general rule of matrix multiplication.
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If π΄ and π΅ are both diagonal matrices, then the two matrices are commutative.
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π΄π΅ is equal to π΅π΄.