Question Video: Finding Unknown Elements of a Symmetric Matrix | Nagwa Question Video: Finding Unknown Elements of a Symmetric Matrix | Nagwa

# Question Video: Finding Unknown Elements of a Symmetric Matrix Mathematics • First Year of Secondary School

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If matrix π΄ = β6, β2π¦, βπ¦ β 3π₯ and π§ + 5, β9, π§ β 6 and β9 β 2π₯, β11, β4, is a symmetric matrix, what are the values of π₯, π¦, and π§?

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### Video Transcript

If π΄ β the matrix shown on screen β is a symmetric matrix, what are the values of π₯, π¦, and π§?

A symmetric matrix is one thatβs equal to its own transpose. π΄ is equal to π΄ transpose. The transpose of a matrix is found by swapping the rows and columns around. The first row of matrix π΄ becomes the first column of the matrix π΄ transpose, the second row of the matrix π΄ is the second column of the matrix π΄ transpose, the third row of the matrix π΄ is the third column of the matrix π΄ transpose.

As π΄ is a symmetric matrix and therefore π΄ is equal to π΄ transpose, this means that every element in π΄ is equal to its corresponding element in π΄ transpose. We can see this clearly for the elements on the diagonal which are just numbers. But itβs also true for the elements of the diagonal, which are mostly expressed in terms of the variables π₯, π¦, and π§. By looking at corresponding elements between the two matrices, we can form some equations, which we can solve in order to find the values of π₯, π¦, and π§.

For example, the element in the second row and third column of each matrix must be equal. Therefore, we have the equation π§ minus six is equal to negative 11. Adding six to both sides of this equation gives the value of π§. Itβs equal to negative five. We would have got the same equation and hence the same value of π§ if weβd instead equated the elements in the third row and second column of the two matrices.

Now, letβs look at what other elements we can equate in order to find the value of either π₯ or π¦. If we equate the elements in the first row and second column of the two matrices, then we have a relationship between π¦ and π§. Negative two π¦ is equal to π§ plus five. This is the same relationship that we would see if we equated the elements in the second row and first column.

We already know that π§ is equal to negative five. So we can substitute this value of π§ into the equation in order to solve for π¦. Negative two π¦ is equal to negative five plus five. Negative two π¦ is equal to zero. And dividing both sides of the equation by negative two, we see then that π¦ is equal to zero.

Next, we need to find an equation that we can use to find the value of π₯. If we equate the elements in the first row and third column of the two matrices, then weβll have a relationship between π¦ and π₯. Negative π¦ minus three π₯ is equal to negative nine minus two π₯. Again, this same relationship could be achieved by equating the elements in the third row and first column. π¦ remember is equal to zero. So substituting this value into the equation gives zero minus three π₯ is equal to negative nine minus two π₯.

To solve this equation, we begin by adding two π₯ to each side. This gives negative π₯ is equal to negative nine. In order to find the value of π₯, we need to divide or multiply both sides of the equation by negative one. This gives π₯ is equal to nine.

The values of π₯, π¦, and π§ found by equating corresponding elements of π΄ and π΄ transpose are π₯ is equal to nine, π¦ is equal to zero, π§ is equal to negative five.

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