Question Video: Understanding the Properties of Inverse Matrices and the Identity Matrix | Nagwa Question Video: Understanding the Properties of Inverse Matrices and the Identity Matrix | Nagwa

Question Video: Understanding the Properties of Inverse Matrices and the Identity Matrix Mathematics • Third Year of Secondary School

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Given the matrices π΄ and π΅, where π΄ = [1, β2, 3 and 0, β1, 4 and 0, 0, 1] and π΅ = [1, β2, 5 and 0, β1, 4 and 0, 0, 1], find π΄π΅. Without doing any further calculations, find π΄β»ΒΉ.

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Video Transcript

Given the matrices π΄ and π΅, where π΄ equals one, negative two, three, zero, negative one, four, zero, zero, one and π΅ equals one, negative two, five, zero, negative one, four, zero, zero, one, find π΄π΅. And the second part of the question says, βWithout doing any further calculations, find π΄ inverse.β

So the first thing weβre going to do here is find the product π΄π΅. Using the usual method for multiplying three-by-three matrices together, we find that π΄π΅ is one, zero, zero, zero, one, zero, zero, zero, one. And we notice that this is actually the three-by-three identity matrix. So what does this mean for our matrices π΄ and π΅?

Well, the definition of the inverse matrix is that itβs the π΄ inverse such that π΄ multiplied by π΄ inverse equals the identity matrix. So the fact that we found the product π΄π΅ to be the identity matrix means that the matrix π΅ must be the inverse of the matrix π΄.

The second part of the question says, βWithout doing any further calculations, find π΄ inverse.β Well, because when we find the product π΄π΅ we get the identity matrix, this means that the matrix π΅ is the inverse of π΄. Therefore, π΄ inverse is the matrix π΅, which is one, negative two, five, zero, negative one, four, zero, zero, one.

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