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If the matrix π΄ is equal to one, zero, zero, one, which of the following is true?
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The matrix π΄ is a row matrix.
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The matrix π΄ is a column matrix.
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The matrix π΄ is an identity matrix.
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Or the matrix π΄ is a zero matrix.
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Letβs consider each of these four statements and remind ourselves of the features of the type of matrix they refer to.
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Weβll start with the final statement: the matrix π΄ is a zero matrix.
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A zero matrix can have any dimensions, but every element within the matrix must be equal to zero.
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Looking at our matrix π΄, we can see that it does indeed have two elements which are equal to zero.
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But the other two elements, the elements which are on what we call the leading diagonal, are not equal to zero.
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So matrix π΄ is not a zero matrix because it doesnβt have every element equal to zero.
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Letβs now consider the top statement: the matrix π΄ is a row matrix.
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A row matrix or row vector is a matrix of order one by π.
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Itβs simply a matrix which has only one row, although it can have any number of columns.
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In its general form, it would look something like this, and the elements would be labeled as π one one, π one two, all the way up to π one π.
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Looking at our matrix π΄, we can see that it has not one but two rows.
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And therefore, the matrix π΄ is not a row matrix.
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Weβll now consider the second statement: the matrix π΄ is a column matrix.
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Well, a column matrix or a column vector is a matrix of order π by one.
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So it can have any number of rows, but it must have only one column.
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In its general form, it would look a little something like this.
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And we could label the elements as π one one, π two one, all the way down to π π one.
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Looking at our matrix π΄, itβs easy to see that this matrix has two columns, not one.
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And therefore, the matrix π΄ is not a column matrix.
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Weβre left with only one possibility.
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Is the matrix π΄ an identity matrix?
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Well, an identity matrix is, first of all, a diagonal matrix which also means it must be square.
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Being a diagonal matrix means that all of the elements not on the leading diagonal, thatβs the diagonal from the top left to the bottom right, must be equal to zero.
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We can see that this is the case for our matrix π΄.
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Furthermore, in order to be an identity matrix, rather than just a diagonal matrix, all of the elements on the leading diagonal must be equal to one.
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Looking at our matrix π΄, we can see that this is true for the two elements on the leading diagonal.
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So we have a diagonal matrix.
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All of the elements not on the leading diagonal are equal to zero.
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And all of the elements that are on the leading diagonal are equal to one, which means that the matrix π΄ is an identity matrix.
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In fact, because itβs a matrix of order two by two, we sometimes refer to it as πΌ two, the two-by-two identity matrix.
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The only statement which is true then is this one: the matrix π΄ is an identity matrix.