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Question Video: Solving a Matrix Equation Involving an Invertible Matrix Mathematics • 10th Grade

True or False: Given the ๐ด is an invertible matrix satisfying the equation ๐ด๐‘‹ = ๐ต, the solution of this matrix equation is ๐‘‹ = ๐ดโปยน๐ต.

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Video Transcript

True or False: Given the ๐ด is an invertible matrix satisfying the equation ๐ด๐‘‹ equals ๐ต, the solution of this matrix equation is ๐‘‹ equals ๐ด inverse ๐ต.

A matrix equation is an equation where the variables stand for matrices. Weโ€™ve been asked if the solution of the matrix equation ๐ด๐‘‹ equals ๐ต is ๐‘‹ equals ๐ด inverse ๐ต. In order to find a solution to this matrix equation, thereโ€™s one important thing to recall, that is, that ๐ด inverse multiplied by ๐ด equals the identity matrix, and thatโ€™s the same as ๐ด multiplied by ๐ด inverse. So if we take ๐ด๐‘‹ equals ๐ต and we multiply on the left by ๐ด inverse, weโ€™ve got ๐ด inverse multiplied by ๐ด multiplied by ๐‘‹ equals ๐ด inverse multiplied by ๐ต. This is okay only because weโ€™re told that ๐ด is an invertible matrix.

Now remember that matrix multiplication is associative. This means that this is the same as ๐ด inverse multiplied by ๐ด multiplied by ๐‘‹ equals ๐ด inverse multiplied by ๐ต. Now by definition, we know that ๐ด inverse multiplied by ๐ด gives us the identity matrix. But we know that taking a matrix and multiplying it by the identity matrix just gives us that original matrix, so we have that ๐‘‹ equals ๐ด inverse ๐ต. And therefore, this statement is true. This is a particularly useful result because we can use this to help us solve systems of linear equations.

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